A Course of Mathematical Analysis 0677201303, 9780677201306

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A. Y. KHINCHIN

A Course of MATHEMATICAL ANALYSIS

A COURSE OF MATHEMATICAL ANALYSIS In the Series

INTERNATIONAL MONOGRAPHS ON ADVANCED MATHEMATICS & PHYSICS

FU R TH E R TITLES IN T H IS SERIES 1. 2.

M ik h l i n :

L inear In teg ral Equations.

K o r o v k in :

L inear O perators an d the T heory of A pproxim ation.

3.

K h in c h in :

A nalytical T heory o f Statistical Physics.

4.

E P s g o lt’s : K r a s n o s e ls k y :

D ifferential Equations.

5.

Convex Functions an d O rlich ’s Spaces.

6.

L a v re n ty e v -^ S h a b a t: M ethods of the T h eory of Functions o f a Com plex V ariable.

7.

L u s te r n ik & S obolev : Elem ents of Functional Analysis.

8.

B lo k h in :

M ethods of X -R ay Spectrum Analysis.

9.

G h e c h u lin :

W ave Processes, O ptics an d Elem ents of A tom ic an d N uclear Physics.

10.

R o m a n o v s k i:

F ourier Series, Field T h eory an d L aplace T ransform .

11.

F u c h s & S h a b a t:

Functions o f a Com plex V ariable an d some of their applications.

12.

G o ld a n s k ii:

C ounting Statistics in R ecording N uclear Particles.

13.

S h to k a lo :

L inear D ifferential Equations w ith V a ri­ able Coefficients.

14.

G ra d s h te in :

D irect and Reverse Theorem s.

15.

N o v o z h ilo v :

E lem entary Particles.

A COURSE OF

MATHEMATICAL ANALYSIS

by

Academician A. K H IN C H IN Moscow University U.S.S. R.

Translated from, the third Russian Edition (1957)

I960

Hindustan Publishing Gorp. (India) DELHI

Published by : HINDUSTAN PUBLISHING CORP. (INDIA) 6, U. B. Jawahar Nagar, DELHI-6.

©

Copyright I960

Copyright reserved by the Hindustan Publishing Ccrp. [India). This book, or parts thereof, may not be reproduced in any form without the written permission o f the publishers.

PRINTED IN INDIA A T THE CENTRAL ELECTRIC PRESS. KAMLA NAGAR, DELHI-6. PHONE : 20|23

PREFACE TO THE FIRST RUSSIAN EDITION T his course o f m athem atical analysis is a text-book for students o f m e ch a n ic o -m a th e m atica la n d physico-m athem atical faculties of our universities (and to some extent of pedagogical institutes as well) ; it is intended as the m ain text-book in the study o f a science w hich appears in the curriculum u n d e r the heading of m athem atical analysis and w hich deals w ith the theory o f limits, infinite series and differential calculus w ith simple applications o f these subjects. T h e necessity for such a text-book arose as most of the text-books on m athem atical analysis published in this country have not fully satisfied the above requirem ents. Text-books w hich by their briefness and sim plicity o f tre a tm e n t are w ithin the reach o f the average student are usually either obsolete or of lower scientific level th an is required for the train in g o f specialized-m athem aticians ; other text-books w hich keep on the m odern level are usually very bulky and their contents reach far beyond the scope o f the cu rren t curriculum so th a t the average first and second years students are unable to benefit by them . It was therefore necessary to w rite a text-book whose contents w ould only include the strict requirem ents of the current curriculum and Which w ould, at the same tim e, fully conform to the m odern scientific stan d ard . In a ttem p tin g to m ake this text-book as brief as possible, I have selected the m inim um necessary m aterial an d avoided all slackness in treatm en t. O n the other hand, to help the stu d en t as far as possible, only the m inim um detail is given throughout this course. I have not used w ords sparingly while trying to explain the line o f argum entation. T h e relatio n sh ip am ong various concepts, theorem s, problem s and theories, th eir im portance an d m ethod o f ap p licatio n in the applied fields and industry, as well as m any other points o f m athem atical analysis are, in m any cases dealt here m ore comprehensively and system atically th a n is usually done in other m ore extensive text-books. I have tried to m ake the student ready to appreciate the in tro d u c­ tion of new concepts a n d construction o f new theories and make him accept them n a tu ra lly an d inevitably. I think th a t it has been only thus possible to m ain tain the continuity of interest o f the student and m ake him absorb the. subject in an inform al m anner. A n experienced read er will probably find th a t the theory of lim its has been discussed in its full detail in chapters II , I I I and IV .

11

T h is theory is traditionally presented to secondary school students on the X V III century level ; university text-books m a th em atical analysis im m ediately give the m odern treatm en t of the theory of limits w ith all the s's and 8's, and this is often preceded by a c h a p te r devoted to the general theory of real num bers—a subject w hich does not, in fact, belong to analysis b u t to the theory of num bers an d the theory of sets.. As a result, the student thinks th a t the new “ u niversity” treatm ent of lim iting processes has n oth in g in com m on w ith those limits which he has known a t school. In the second place— a n d th is is even m ore im p o rtan t—this m ethod can rob the stu d en t o f elem ents of m athem atical analysis as a live, dynam ic and dialectic science w hich find their place in the history of scientific developm ent a n d w hich have even today m any of their p ractical applications. T h e se undesirable effects w hich I h ad occasion to observe in m any instances during my career as a teacher prom pted m e to use in this text a com ­ pletely new system for treating the theory of limits. T his system essen­ tially involves the following. A t first (chapter II) the theory of lim its is m ainly based on an elem entary b u t not a com pletely form al basis, and concepts like “ process” o r“ m om ents” w hich are not fully defined anyw here are system atically used. O nly afterw ards the necessity of form alisation has been em phasised, and the fu n d am en tal m ath em atical types of processes defined (c h a p ter I I I ) . T h e n th e atten tio n o f th e student is draw n to the necessity of constructing a general theory of real num bers, and such a theory is, in fact, given in ch ap er IV . T his m ethod of treatm en t, w hich I h a d occasion to test three tim es in practice has the useful advantage th a t it creates in the m ind of the student a gradual transition from the “ school” theory of limits to its “ university” treatm ent, an d all stages o f this tre a tm e n t, are fully explained. At the sam e tim e it enables him to create a t the begin­ ning and m aintain throughout the course the basic concepts of m athem atical analysis as a live and dynam ic subject and co n cen trate on the formally logical refinem ents of this subject, w hich is its due. So far as the general theory of real num bers is concerned, I have found it necessary to convince the read er o f its significance and quote one of the possible principles explaining the existence o f im aginary num bers (the lim it of a m onotone bounded sequence). O nly then I have enum erated the basic problem s w hich the theory has to deal (order in a continuum , definitions an d rules of algebraic methods) ; at this point I have also given few exam ples of their solution, indicating briefly th a t the theory of num bers can be applied quite satisfactorily to . these problem s and th a t, in fu tu re, we shall

Ill

deliberately use the results provided by this theory. T h e future m athem aticians will be able to learn in o th er m ore detailed courses the fact th a t the theory of num bers can solve all these p ro b lem s; this problem is h ard ly of any interest to the future m echanic, physicist, or astronom er. In any case, I do not consider it possible to a ttra c t the a tte n tio n of a varied audience, either in m y lectures or in this book, to the study of a large ch ap ter the contents of w hich have no im m ediate connection w ith m athem atical analysis. T h e fu rth er tre a tm e n t of the subject follows, in its m ain o u t­ lines, certain well defined m ethods I am sorry to say th a t in editing the last three chapters (m ultiple, curvilinear an d surface integrals) m y attem p ts to m ake the treatm en t absolutely form al an d , a t the sam e tim e, easily accessible did not m eet w ith success, as far as I am able to judge. I could not avoid compromises by sacrificing either the form ality or the briefness and accessibility o f the argum ents. If this course is received w ith favour, then it will undoubtedly be necessary to work fu rth er on these chapters in future editions. A few problem s given in this course are valuable only as illustrations, b ut they are not intended as a m ethod of instruction. T h e num ber an d ch aracter of these problem s correspond to w hat a lecturer can convey during his lectures. I h ad no intention of including the m aterial for practical (group) lessons in this course of analysis. O bviously, anyone studying this book should sim ultan­ eously use a good book for problem s. For this purpose the recently published “ P roblem Book on M athem atical Analysis” by B.P. D em idovich (G ostekhizdat, 1952) is p articu larly suitable. For the convenience o f certain classes of readers I have indicated in m any p aragraphs a few problem s a p p earin g in the above book which I especially recom m end. I m ust, how ever, w arn the reader th a t these problem s are, in a m ajority of cases, insufficient for acquiring the necessary skill ; a fu rth er choice of examples should be left to ihc teacher-in-charge of practical classes. A com petent read er will readily note th a t the o rd er in w hich individual subjects are treated in this book is in no w ay com pulsory an d can be easily altered in m any instances ; for exam ple 1) some geom etrical applications of differential calculus (chapter X X I I I ) can be given (and are, in fact, usually given) m uch earlier, and 2) the integral test for convergence of series m ust not be postponed until the theory of generalised integrals is dealt w ith (chapter X X V ); b u t it can be given w ith the tre a tm e n t of series o f constant signs (chapter X V III , § 68).

I t is my pleasant duty to express my sincere an d deep g ra titu d e to m y colleagues of the F aculty of M ath em atical Analysis at th e Moscow, L eningrad and K iev Universities for their valuable help given by reading the m anuscript (or its individual chapters) an d for their rem arks an d suggestions which have mostly led to significant im provem ent in the treatm en t of the subject. In this respecf I am particularly grateful to Prof. L A. T u m a rin (Moscow) an d Prof. G.E. Shilov (K iev). Finally, I w ant to th an k the editor of my book, O .N . Golovin, for his com petent an d considerable work devoted to this book ; his m any valuable suggestions have h elp ed considerably to im prove its contents. M oscow, 24 February, 1953.

A. K H IN C H IN

PREFACE TO THE SECOND RUSSIAN EDITION T h e second edition of this book is m ainly printed from blocks^ an d corrections of m any individual mistakes an d errors have been done by the a u th o r; in some cases attem pts have been m ade to im prove the tre a tm e n t of the subject. In this respect I have been greatly helped by a detailed criticism of this book sent to m e by the Faculty of M ath em atical Analysis a t the Rostov U niversity (under the chairm anship of Prof. F.D . G ak h o v ); I am deeply grateful to all the m em bers of this F aculty. I am also very thankful to A cadem i­ cian A .N . K olm ogorov an d Prof. A.D. M yshkis (M insk) for pointing out some mistakes. T h e “ Problem Book on M athem atical A nalysis” by B.P. D em idovich w hich has been frequently referred in this book has ap p eared in its second edition in 1954 w ith a fundam ental revision of the n um bering o f problem s. In the present edition of this course the num b erin g o f all recom m ended exercises refer to the first edition of this “ Problem Book.” M oscow, 19 December, 1954.

A. K H IN C H IN

CONTENTS C h a p te r 1. F U N C T IO N S § 1. V ariables § 2. Functions § 3. T h e region of definition of a function § 4. Functions an d form ulae § 5. The geom etrical representation of functions § 6. E lem entary functions

... ... ... ... ... ...

C h ap ter 2. E L E M E N T A R Y T H E O R Y O F L IM IT S ... § 7. Infinitesim al quantities ... § 8. O perations w ith infinitesim al quantities § 9. Infinitely large quantities ... § 10. Q uantities w hich tend to lim its ... § 11. O perations w ith quantities w hich tend to limits ... § 12. Infinitesim al and infinitely large quantities of different orders. ... C h a p te r 3. T H E D E V E L O P M E N T O F T H E A C C U R A T E T H E O R Y O F L I M I T T R A N S IT IO N § 13. T h e m ath em atical definition of a process § 14. T h e accurate concept of lim its § 15. T he developm ent of the concept of lim it transi­ tions

i 1 3 6 7 11 13 18 18 23 26 29 33 39

... ... ...

45 45 47

...

52

C h a p te r 4. R E A L N U M B E R S ... § 16. Necessity of producing a general theory of real num bers ... § 17. C onstruction of a continuum ... / § 18. F u n d a m e n ta l lem m as ... § 19. F inal points in connection w ith the theory of limits ...

56

74

C h a p te r 5. § 20. § 21. § 22. § 23. § 24.

79 79 84 85 87 94

C O N T IN U O U S F U N C T IO N S D efinition of continuity O perations w ith continuous functions C ontinuity of a com posite function F u n d a m e n ta l properties of continuous functions C ontinuity of elem entary functions

... ... ... ... ... ...

56 59 69

viii C h apter 6. § 25. § 26. § 27. § 28. § 29. § 30.

D E R IV A T IV E S ... U niform and non-uniform variation of functions ... Instantaneous velocity of non-uniform m o v e m e n t... Local density o f a heterogeneous rod ... Definition of a derivative ... Laws of differentiation ... T he existence of functions and th eir geom etrical illustration ...

98 98 101 106 108 110

C hapter 7. D IF F E R E N T IA L S ... § 31. Definition and relationship w ith derivatives ... § 32. G eom etrical illustration and laws for evaluation ... § 33. In v a ria n t ch aracter of the relationship betw een a derivative and a differential ...

128 128 132

C hapter 8. § 34. § 35.

123

134

D E R IV A T IV E S AND D IF F E R E N T IA L S O F H IG H E R O R D E R S ... 136 Derivatives of higher orders ... 136 Differentials of higher orders an d their relatio n ­ ship w ith derivatives ... 139

C h ap ter 9. M EA N V A L U E T H E O R E M S ... § 36. T heorem on finite increm ents ... § 37. Evaluation of limits of ratios o f infinitely sm all and infinitely large quantities ... § 38. T a y lo r’s form ula ... § 39. T he last term in T ay lo r’s form ula ...

142 142 147 154 158

C hapter 10. § 40. § 41.

APPLICATION^ O F D IF F E R E N T IA L C A L ­ C U L U S T O ANALYSIS O F F U N C T IO N S ... 164 Increasing and decreasing o f functions ... 164 E xtrem a ... 167

C hapter 11. IN V E R S E O F D IF F E R E N T IA T IO N § 42. C oncept of prim itives § 43. Simple general m ethods o f in teg ratio n

... 175 ... 175 ... 182

C hapter 12. IN T E G R A L § 44. A rea o f a curvilinear trapeziu m § 45. W ork of a variable force § 46. G eneral concept of an integral § 47. U p p er and lower sums § 48. Integreability of functions

... 193 ... 193 ... 198 ... 201 ... 204 ... 207

IX

C h ap ter 13. § § §

R E L A T IO N S H IP B ET W E EN AN IN T E G ­ R A L A N D A P R IM IT IV E ... 213 49. Sim ple properties of integrals ... 213 50. R elationship betw een an integral an d a prim itive ... 218 51. F u rth e r properties of integrals ... 223

C h ap ter 14. § § § §

T H E G E O M E T R IC A L A N D M E C H A N I­ CAL A P P L IC A T IO N S O F IN T E G R A L S ... 52. L ength of an arc of a plane curve ... 53. L engths of arcs of curves in space ... 54. M ass, centre of grav ity and m om ents of inertia of a m aterial plane curve ... 55. C apacities of geom etrical bodies ...

C h ap ter 15. § § §

A P P R O X IM A T E E V A L U A T IO N TEGRALS 56. Problem atic set up 57. M ethod o f trapezium s 58. M ethod of parabolas

230 230 241 242 247

O F IN ­

C h ap ter 16. § § §

IN T E G R A T IO N O F R A T IO N A L F U N C ­ T IO N S 59. A lgebraical in troduction 60. In teg ratio n of sim ple fractions 61. O strogradskij’s m ethod

... 254 ... 254 ... 257 ... 262 ... 265 ... 265 ... 274 ... 277

C h ap ter 17.

§ 62.

§ 63.

IN T E G R A T IO N O F T H E S IM P L E R A ­ T IO N A L AND T R A N SC E N D E N T A L F U N C T IO N S ... In te g ra tio n of functions of the type

*('•vs?)

282

In teg ratio n of functions of the type 72 (x} y/ax*-rbx -r c)

§ 64. § 65. § 66.

282

... ... ...

Prim itives of binom ial differentials In te g ra tio n o f trigonom etrical differentials In teg ratio n o f differentials containing exponential functions ... C h a p te r 18. N U M E R IC A L IN F IN IT E SE R IE S ... § 67. F un d am en tal concepts ... § 68. Series w ith constant signs ... § 69. Series w ith variable signs ... § 70. O perations w ith series ... § 71. Infinite products •••

284 287 289 294 297 297 305 316 320 326

X

C hapter 19. IN F IN IT E SE R IE S O F F U N C T IO N S ' ... § 72. R egion of convergence of a series o f functions ... § 73. U niform convergence ... § 74. T he continuity of the sum of a functional series ... § 75. T erm -by-term integration and differentiation of series ••• C hapter 20 P O W E R SE R IE S A N D SE R IE S O F P O L Y ­ N O M IA L S ... § 76 R egion of convergence of a pow er series § 77. U niform convergence and its consequences § 78. Expansion of functions into pow er series § 79. Series of polynom ials § 80. T heorem of W eierstrass C hapter 21. T R IG O N O M E T R IC A L SE R IE S § 81. Fourier coefficients § 82 A verage approxim ation § 83. D irichlet^Liapunov theorem on closed trig o n o ­ m etrical systems Convergence of Fourier series 84. § § 85. G eneralised trigonom etrical series C h ap ter 22. D IF F E R E N T IA T IO N O F F U N C T IO N S O F SE V ER A L V A RIA BLES § 86. C ontinuity of functions o f several independent variables § 87. Tw o-dim ensional continuum § 88. Properties of continuous functions § 89. P artial derivatives § 90. D ifferentials § 91. Derivatives in arb itra ry directions § 92. D ifferentiation of com posite an d im plicit fu n c­ tions 93. Hom ogeneous functions an d Euler theorem § § 94. P artial derivatives of higher orders . § 95. T a y lo r’s form ula for functions of two variables .§ 96. E xtrem a C h ap ter 23. SO M E S IM P L E G E O M E T R IC A L A P P L I­ C A T IO N S O F D IF F E R E N T IA L C A L C U L U S ... § 97- E quations of tan g en t an d norm al to a plane curve ... ; ;§ 98. T angential line an d norm al plane to a curve in , *, space

333 333 335 340 344 351 351 357 361 369 372 377 377 383 388 394 396 400 400 403 408 410 413 419 422 427 429 433 438 443 443 446

XI

§ § § §

99. 100. 101. 102.

T an g en tial an d norm al planes to a surface D irection o f convexity an d concavity o f a curve C urvature o f a plane curve T an g en tial circle

448 451 453 453

...

C h a p te r 24. IM P L IC IT F U N C T IO N S § 103. T h e sim plest problem § 104. T h e general problem § 105. O strogradskij’s determ inant § 106. C onditional extrem um

462 462 469 475 483

C h ap ter 25. G E N E R A L IS E D IN T E G R A L S § 107. Integrals w ith infinite limits §108. Integrals o f unbounded functions

491 491 504

C h ap ter 26. IN T E G R A L S O F P A R A M E T R IC F U N C ­ T IO N S § 109. Integrals w ith finite limits § 110. Integrals w ith infinite limits § 111. Exam ples §112. E u ler’s integrals § 113. S tirling’s form ula

514 514 526 535541 548

C h ap ter 27. D O U B L E AND T R IP L E IN T E G R A L S § 114. M easurable plane figures § 115. V olum es of cylindrical bodies § 116. D ouble integral § 117. E valuation of double integrals by m eans of two simple integrations § 118. S ubstitution o f variables in double integrals § 119- T riple integrals § 120. A pplications

557 557 567 571 576, .584590' 593

C h a p te r 28. C U R V IL IN E A R IN T E G R A L S . § 121. D efinition of a plane curvilinear integral . . § 122. W ork of a plane field of force . § 123. G reen’s form ula § 124. A pplication to differentials of functions of two . variables . § 125. C urvilinear integrals in space

602 602 610' 612

C h a p te r 29. S U R F A C E IN T E G R A L S § 126. T h e simplest case § 127. G eneral definition of surface integrals

626' 626630^

. .

617 622'

xii § 128. O strogradskij’s form ula § 129. Stoke’s form ula § 130. Elem ents of the field theory

... ... ...

637 642 647

C O N C L U S IO N — Short historical sketch

...

653

...

665

fIN D E X

CHAPTER I FU N C T IO N S § 1.

Variables

The introduction o f the variable was a decisive step in mathematics. Thus movement and dialectics were introduced in mathematics. (F. Engels, Dialectics of Nature, Gospolizdat, 1948, p. 208.) Elementary mathematics—the mathematics of constants— revolves, as it were, within limits of formal logics ; the mathematics of variables, which is chiefly concerned with infinitely small quantities, essentially involves the application of dialectics to mathematical relationships. (F. Engels, AntiDuring, Gospolizdat„ 1948, p. 127.)

W hen we observe a n a tu ra l phenom enon or the course of a technical process we can usually note the different behaviour o f quantities involved in this phenom enon or process. Some quantities do not change in the course of the process, i.e. they rem ain “ constant” , while others are subjected to greater or lesser change— they becom e g re a ter or sm aller— i.e. they are “ v ariab le” . I f we heat a gas con­ fined in a closed vessel its volum e rem ains constant ; the n u m ber o f molecules of the gas also rem ains c o n sta n t; on the other h and the tem pe­ ra tu re of the gas, and its pressure will grow and becom e increasingly greater. T he p ictu re becomes even m ore varied if instead of considering this simple laboratory experim ent we consider a com plicated techni­ cal process. Let us consider, for exam ple, the flight o f an aeroplane. M an y different quantities are involved in this phenom enon. Some o f these rem ain constant throughout the flight ; e.g. the num ber of passengers, the w eight of their luggage, the span of the wings of th e aeroplane, and- m any others. H ow ever, this process also involves m a n y other quantities which alter during the process by becom ing g re a ter or smaller. Such are, for exam ple, the distance of the aeroplane from the point o f d ep artu re and from its destination, its height above th e earth, the supply of fuel, the tem peratu re, pressure and hum idity

9

A C O U R S E O F M A T H E M A T IC A L A N A L Y SIS

o f the surrounding air, and m any others. T h e above sum m ary shows th at these variable quantities are most im p o rtan t in econom ical an d technical calculations connected w ith this process. T his can readily be understood. N ature involves continuous changes an d the p ractical life of m an is directed tow ards changing his surroundings. For this reason processes in w hich nothing, or alm ost nothing, changes have little to offer scientifically and are of no practical interest. A ccording to the dialectic principles of n atu re study, we should study not so m u ch the instantaneous aspect o f phenom ena b u t th eir changes in tim e ; from the dialectic point of view we are not so m uch interested in the given aspect of a phenom enon b u t in the general course of the ph en o ­ m enon, i.e. we are interested how and w hat changes if this p h en o ­ m enon took place from tim e to tim e. M athem atics, in as far as it is a real tool in n atu re study, should be able to provide a n a p p aratu s w hich would enable one to study system atically any changes in quantities w hich take place in n ature an d in technical processes. M athem atical analysis is such an ap p aratu s an d , in the widest sense of the w ord, can be called the m ath em atical science o f variables. Hence the first basic concept in m ath em atical analysis is the variable qu an tity or, as it is usually said in m athem atics, the concept o f the variable. By this we m ean quantities w hich acquire varying values,' either greater or smaller, in the course o f the given] process ; at different stages of a given process the values o f this q u an tity are, generally speaking, different. W ithout going into fu rth er details we know from everyday experience th a t the ch aracter an d m an n er in w hich quantities change can follow a very diverse course ; some quantities increase continuously; other quantities, on the o th er h an d , decrease continuously; still others change in a v ib ratin g m an n er by first growing an d then dim inishing (the distance of the E a rth from the Sun, the deflection o f a pendulum from the vertical position) ; if we assume th a t the given qu an tity grows continuously, it can do so either very rapidly or very slowly, i.e., the pace of its grow th can becom e quicker or slower. M athem atical analysis in its w idest sense en­ ables us to study systematically these a n d o th er characteristic changes o f quantities in our su rro u n d in g s; it introduces a definite p a tte rn into the enorm ous num ber of various types o f changes an d finds com m on laws which govern changes of various types. In m athem atics every q u an tity involved in a phenom enon, irrespective of w hether it is a constant or a variable, is usually denoted by a single letter. T hus, for exam ple, if a q u an tity is denoted by the letter x or by the letter a, then this fact by itself gives no indication as

F U N C T IO N S

3

to w hether this q u a n tity is a constant or a variable ; therefore the *way in w hich this quantity changes m ust be stressed separately. F u rth erm o re it is very im p o rtan t to keep in m ind the fact th a t w ith o u t the know ledge o f the process (phenomenon) in hand, we cannot, gener-ally speaking, know w hether this or ano th er q u an tity is a constant or a variable. T h e same q u an tity can be a constant in one process an d -a v aria b le in an o th er process ; thus, for exam ple, if we ro tate a circle o f radius r ab o u t a straight line w ithout changing its radius (first process) then the area of this circle tty2 will be c o n s ta n t; if, however, *we keep the centre of the circle stationary an d increase its radius (second process), then the area of the circle will grow, i.e. it will be ^ variable. In m ath em atical analysis the well-known geom etrical represen­ ta tio n of num bers by points on a straight line (the so-called “ n u m ber lin e ’5) is w idely used. Ifw e denote the origin by 0 a n d a u n it of length on the straight line, then we can represent an a rb itra ry n u m b er a by a p o in t a t a d istance | a | *from the point 0 ) in a direction w hich d epends on the sign of the nu m b er a. (generally, if the n u m b er line is h o rizo n tal, positive num bers are plotted to the right and negative n u m b ers to the left o f the p o in t 0 ). Every value o f a: is a n u m b er a n d can be represented by a point on the n u m b er line. I f in the given process the value of x is constant then this value is denoted by o n e an d the sam e point on the nu m b er line during the whole process, W e can therefore say th a t a constant is represented by a stationary p o in t on the num ber line. If, how ever, the value o f x varies during th e given process, then its values a t different stages o f the process are represented by different points on the n u m b er lin e ; in the course o f th e process the p oint denoting the value o f x changes its position an d w e can therefore say th a t a variable is denoted by a mobile p oint on th e n u m b er line. § 2. Functions Q u an tities involved in the same phenom enon do not, as a rule, c h an g e independently o-f each o th e r; usually these quantities are also m ore or less closely related to one an o th er so th a t changes in one o f these quantities involve corresponding changes in the other quantities. T h u s, by increasing the radius of a circle we inevitably also increase its a r e a ; by compressing a gas confined in a vessel {i.e. by decreasing th e volum e occupied by the gas) we also (by keeping the tem p eratu re constant) inevitably increase the pressure o f the gas ; by adding *) The symbol I x j denotes the “ absolute value of the number x.”

4

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

m anure to the soil we hope to increase the yield o f the harvest, etc. W e can see from the above examples th a t quantities involved in th e same phenom enon can b e ar to one anoth er a more or less close re ­ lationship. T his relationship is closest in the first ex am p le ; by know ing the radius r of the circle we can determ ine its area s un iq u ely a n d w ith absolute accuracy according to the form ula 0). W hen a is a ratio n al fraction, eg. a = p\q (w h erep an d q are integers a n d we can always assum e th a t q > 0) then v Xa —

_

X q = % JxT *

is an irrational algebraic junction o f x (for the operations perform ed over x include the extraction of a root o f an a rb itra ry degree q).

F U N C T IO N S

15

T h e values of this function can no longer be evaluated as simply as those of a ratio n al function. T his is even m ore tru e in the case when a is a n irratio n al nu m b er (i.e. for exam ple the function y — x ^ “ or y — x ^ ) \ strictly speaking we do not know how to determ ine such fu n c tio n s; we shall re tu rn to this question in §§ 17 an d 24. T he region of definition of the function given by the form ula y — x a depends on the n a tu re of the n u m b er a. I f a is a positive integer then the whole num ber line serves as its region of d efin itio n ; b u t w hen a is a negative integer or zero, i.e. a ^ 0, the point x = 0 m u st be excluded from this line. I f a = l/q, w here q is a positive integer, then the function will be determ ined for all values of x w hen q is odd, and only for x ^ 0, w hen q is even. T h e reader will be able to determ ine for him self the region of definition of the fu n ction x a w hen a = pjq, w here p an d q are integers. In cases where a is irratio n al, its region of definition is the sem i-straight line x > 0, as we shall learn in § 174. Exponential functions. is know n

By this nam e the following function

y = a x, w here a is a constant positive num ber. W e shall learn in §17 th at th e whole nu m b er line serves as the region of definition of this function. W e shall learn later some other im p o rtan t properties o f this function. T h e value o fy for the. given value of x cannot, in this case, be obtained by m eans of any know n finite sequence of opera­ tions (with the exception of the trivial case w hen a = 1 ); the func­ tio n a 1 is not a n algebraic function b u t a transcendental function *).

*) S tr ictly sp ea k in g th e p r o b le m is as fo llo w s : I f th e fu n c tio n ^ = f (x) o f th e in d e p e n d en t v a ria b le x is o b ta in e d

after p e r fo r m in g a fin ite n u m b e r o f a lg e b r a ic

o p era tio n s, as p ro v ed in a lg e b r a , th e n a p o ly n o m ia l P {x}y) o f tw o v a ria b le s exists so th a t, id e n tic a lly {i.e. for a n y x) P [x , f (x )] = 0 .

T h e co n v er se p ro p o sitio n is n ot

t r u e ; it m a y h a p p e n th a t th e p o ly n o m ia l P do.es e x ist, b u t th e fu n ctio n f (x) ca n n o t b e ex p ressed in term s o f x b y m ea n s o f a

fin ite n u m b e r o f a lg e b r a ic

o p era tio n s.

I t is c u sto m a r y to c a ll th e fu n c tio n f (x) a n algebraic fu n c tio n if a p o ly n o m ia l P ex ists for this fu n c tio n w h ic h possesses th e p ro p erties m e n tio n e d a b o v e . class o f a lg e b r a ic fu n c tio n ^expressed in term s o f

is w id e r th a n

a fin ite

num ber

th e class o f o f a lg e b r a ic

.a lg eb ra ic fu n ctio n is k n o w n as a transcendental fu n ctio n . (for e v e ry a > 0 , a c e n d e n t a l fu n ctio n s.

H e n c e th e

fu n ctio n s w h ich o p era tio n s.

can b e

E v e ry n o n -

T h e fu n ctio n s a x, lo g a x

1), sin x , cos x, a rc sin x , a rc cos x, e tc . arc e x a m p le s o f tran s­

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

16 5.

Logarithmic functions.

T h e function

y = log a*, w here a is a constant positive n u m b er other th a n unity, is defined as the inverse of the exponential function. T his m eans th a t it follows fromjy = lo g a* th a t * = a v. T o be m ore exact this m eans th a t for every x > 0, a single n u m b e ry exists w hich satisfies the relationship a y = * ; this n u m b e ry is know n as the lo garithm o f x of the base (or to the base) a and is denoted as lo g a*. Like the exponential function the logarithm ic function is also a transcendental fu n c tio n ; a p a rt from its great theoretical im portance it is also very im p o rtan t in calcula­ tions; its significance is m ainly due to the basic pro p erty o f this fu nction: lo g a (a [3) = lo g a a + lo g a (3. T h e region of definition o f a logarithm ic function of any base is the sem i-straight line x > 0. 6. Simpler trigonometrical functions. These functions a re the following functions w hich are well-known from the school course o f trigonom etry. y = sin*,

y = cos*,

y = ta n * ,

y = cot*,

y = sec*,

y = cosec*.

T h e chief property of these functions is their periodicity, tan * an d cot * have a period tz and the rem aining four functions a period 2 it. T h e whole num ber line serves as the region of definition of the func­ tions sin * an d cos * ; the functions ta n * an d sec * are defined every­ w here except a t points of the type y = (k + t )

an d the functions cot x and cosec*— everywhere except a t points o f the type y

=

}z i t .

w here h in both cases denotes an arb itra ry integer. 7. Inverse trigonometrical functions. G enerally speaking, the function a (*) is said to be the inverse o f the given function / ( * ) i f it follows from y = a (*)' th a t * = f {y) > W e have seen already th a t the function lo g a * is the inverse of the function a x. In this case the inverse function is unique. H ow ever it is quite possiblefor a given function to have several inverse fu n ctio n s; thus the fu n c­ tion * 2 evidently has the following inverse functions : + \ / x a n d ’ — \ / x , for it follows equally from y = + x an d from y = — y ' *• t h a t * = y 2. I t is a well-known fact th a t each one of the sim pler trigonom etrical functions has an infinite n u m b er o f inverse fu n ctio n s;;

F U N C T IO N S

17

these functions are know n as inverse trigonometrical functions. L et us consider, for exam ple, the fam ily of functions inverse to the sine. I f a is an a rb itra ry nu m b er confined betw een — 1 an d + ], then an infinite n u m b e r of values of x exists for w hich sin x = a ; in p articu lar one such value of jv can be found betw een — ttI2 an d f it is denoted by sin-1 a, so th a t ---- ^ sm r1 a 0, the q u an tity (1 + a) n is an infinitely large q u an tity w hen ft —> go . L et us now consider the operation w ith infinitely large q u a n ­ tities. T h e sum of two infinitely large quantities need not necessarily be an infinitely large q uantity, as can be seen from the follow ing' sim ple exam ple : if .v is a n infinitely large quan tity , then as we saw,. — .v will also be an infinitely large qu an tity ; the sum of these twoquantities is alw ays equal to zero, i.e. it is an infinitesim al q u an tity . W e have, how ever, the following im p o rtan t theorem s : Theorem 1. The sum o f two quantities, one o f which is infinitely large and the other is a limited quantity, is an infinitely large quantity. . Proof. L et us assume th a t in the given process * is infinitely large an d y is a lim ited q u an tity . A positive num ber C exists w hich is such th a t from a certain m om ent of our process onw ards we always have | y | < C; let A b e a n a rb itra ry positive n u m b e r; owing to the fact th a t x is infinitely large, there will be an o th er m om ent in our process-

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

28

after which we always have | x | > A 4- C. H ence by choosing the la tte r of the two m om ents, we shall always h av e after th a t in sta n t: | * ] > A -f C,

| y | < C,

hence *) | .v -f y I ^ | * | — Iy I > -4 + C — C = A. O w ing to the fact th a t the num b er A is arb itra rily large, this proves th a t x -f y is an infinitely large qu an tity . If, as we saw above, the addition of infinitely large q u an tities does not always lead to infinitely large quantities, th en on the o th er hand, the m ultiplication of infinitely large quantities follows the sam e rule as the m ultiplication of infinitely small quantities. Theorem 2. The product o f two infinitely large quantities is an infinitely large quantity. Proof. T he reader is by now w ell-acquainted w ith the a rg u ­ m ents used for proving theorem s of this kind ; we can, therefore, tre a t the proof m ore briefly. If xx an d xz are infinitely g reat in the given process and if A is an a rb itra ry positive num ber, then from a certain m om ent of the process onw ards |

| > V A, a n d from an o th er

m om ent onw ards | x 2 ] > \S A ; b u t from the la tte r of the two m om ents onw ards | x 1 x 2 | = I * i | . | x 2 | > A, w hich proves the theorem . From this, in the sam e w ay as w ith infinitesim al quantities, we o btain w ith the aid of induction : Corollary. The product o f an arbitrary constant number o f infinitely large quantities is an infinitely large quantity. T h e following proposition connects the concept of an infinitely large quantity w ith the concept of an infinitely small q u a n tity : Theorem 3. I f x is an infinitesimal quantity which is never zero, then 1jx is an injinitely large quantity, conversely, i f x is an infinitely large quantity which is never zero, then 1jx is an infinitesimal quantity. T o prove this it is sufficient to note th a t the inequality ] a" ) < s: is equivalent to the inequality 1/| * | > 1/e, an d if the n u m b er e is as sm all as we please, then the n u m b er 1jz is as large as we please. *) W e are u sin g h ere a ru le w h ic h is w e ll-k n o w n in e le m e n ta r y a lg e b r a : the

absolute vatue of a sum is not tess than the difference of the absolute values of its terms.

E L E M E N T A R Y T H E O R Y O F L IM IT S § 10.

29'

Q uantities which tend to Limits

In the above sections we have dealt w ith some of the simple types of changes in quantities, i.e. we have considered quantities w hich decreased indefinitely an d other quantities w hich increased indefinitely an d w hich are know n as infinitesim al an d infinitely large q u antities respectively. Following o u r scheme, we shall now consider the next large class o f a type of change and in doings, so we shall find the concept o f infinitesim al quantities very useful. In practice and in n a tu ra l phenom ena it happens frequently th a t the variable q u a n tity x tends to come infinitely close to a certain constant a, so th a t in the course of the process the absolute value o f ’ the difference betw een these quantities becomes infinitesim al; in such cases it is said th a t the q u an tity x has a limit a in the given process or th a t it tends to a. T his is denoted as follows : lim x — a. the two forms of no tatio n are equivalent. T he w ord lim is m ad e up of the first three letters of the latin w ord limes which means limits or b o undary ; b ut the w ord should be read in English, i.e. “ lim it” . It is obvious th a t the q u an tity x can n o t, in this case, have twodifferent lim its : in fact, if x -> a a and x -> a 2, then the absolute values o f the quantities * — a (or x —►a), w here a m ust be a constant, m eans th a t the absolute value o f the difference x — T ,

or

T2

T .

Exam ple 2. A coin is throw n n times in succession and after each throw it is noted w hether the head or tail turns upw ards. Let us assume th a t after the coin is throw n n times, the head appears m times a t the top ;as n increases m, increases as well. Experience shows th a t w hen the coin is geom etrically regular an d physically hom o­ geneous, then, provided it is throw n a great m any times, the head will ap p ear a t the top h a lf the num ber o f times, i.e. the relationship .mjn tends to 1/2 ; we can take it to be proved em pirically th a t the .absolute value of the difference n

2

w hen n increases indefinitely (by becom ing both positive an d negative), rem ains in the end as small as we please, i.e. this difference becomes •an infinitesimal q u an tity as the n u m b er o f times the coin is throw n increases indefinitely. T herefore in our process m n

1

Exam ple 3. I f the quantity * in a certain process is infinitesim al, then the q u a n tity y = a + bx + c x 2, w here a, b an d c are constants, •tends to the lim it a in this process. In fact, y — a = bx + c x 2 and df x is infinitesimal, then the theorem in § 8 enables us to say th a t the •q u an tity bx-\-cx2 is also infinitesim al. Exam ple 4. If in a certain process the q u an tity * is infinitesim al, 'then cos x tends to unity as its lim it. In fact, from a certain m om ent •of the process onw ards | at ] 0. It follows from the relationship 1 — co s2 * = s in 2.* •that 0 ^ 1

— cos*

.and owing to the fact th a t a p a rt from *, sin * is also an infiniteism al q u an tity (exam ple 4, § 7), it follows th a t sin2 * (corollary 4 o f theorem -2, § 8) and the q u an tity 1 cos* are also infinitesim al an d a re

E L E M E N T A R Y T H E O R Y O F L IM IT S

31

‘Confined betw een zero and the infinitesim al value s in 2 x ; b u t in o u r process this m eans th at ,

COS

Example 5.

X

1 .

L et us prove th a t for every constant a > 0 the

•quantity V a tends to unity as its lim it, provided n increases indefinitely. In fact let it be given arb itrarily th a t £ > 0 ; we know (exam ple 3, § 7) th a t as n increases indefinitely, the q u an tity (1 — s) n is an infinitesim al q u an tity an d the q u an tity (1 + e) n is an infinitely large q u a n tity ; therefore provided n is sufficiently large, we have (1 — t ) n < a < (1 + s) " , hence ■or

1 — s < %/a < 1 + s , \/a — 1| < s ,

w hich proves ou r proposition. T h e above exam ples show us th a t the w ay in w hich a variable ^quantity tends to its lim it can be very diverse in character. T hus in exam ple 1 the tem p eratu re T x tends to its lim it T by decreasing continuously; On the other h an d , the tem p eratu re T 2 (in the same exam ple) tends to this sam e lim it T by increasing continuously. In •example 2 (the experim ent w ith throw ing a coin) theory an d practice show us th a t by increasing the nu m b er of times the coin is throw n, the “ fraction of h ead s” m\n becomes g reater an d sm aller, (and sometimes equal to) 1/2; we are dealing here w ith a q u an tity w hich increases .and decreases in the process u n d er consideration while it tends tow ards its lim it. In spite of the fact th a t quantities show great differences in behaviour when tending tow ards their lim it, they also share m any properties in c o m m o n ; this makes it possible to include them in the -same class. W e shall now study some of their properties. Theorem 1. A quantity which tends to a limit in a given process is a .limited quantity in this process. Proof. L et us assum e th a t in a certain process x a. In this c ase the difference * — a is infinitesim al an d , therefore, from a

32

A C O U R S E O F M A T H E M A T IC A L ANALYSIS-

certain m om ent of the process onw ards \ x — a J < 1; hence ow ing to the fact th a t x = a + {x — a), we have | * | ^ | a | -f- | a: — a | < | a | -r 1 This inequality, on the right-hand side of w hich stands a* certain constant positive num ber, is satisfied from a certain m om ent o f the process onw ards; b u t this m eans th a t the q u an tity x islim ited in this process. Theorem 2. I f in a certain process x -> a and a > o, then from' a certain moment o f the process onwards we shall always have x > o. In other words, if a given qu an tity has a positive lim it, then from a certain m om ent o f the process onw ards the q u a n tity itself will be positive. Proof. L et b be an a rb itra ry positive n u m ber sm aller th a o a (0 < b < a). O w ing to the fact th a t the difference x — a is infinitesim al, we shall have from a certain m om ent o f the processonw ards \x — a | < b ; since x = a + (x — a), we have from th a t m om ent onw ards x ^ a —|x — a| > a — b > 0 , w hich h a d to be proved. Corollary 1. I f in a certain process x —>- a and a < 0, then from a" certain moment o f the process onwards we shall always have x < 0. Corollary 2. I f x —> a, and from a certain moment o f the process onwards x ^ 0, then < 2 ^ 0 . I f from a certain moment o f the process onwards x ^ 0, th en aA, x2 —»- a2, ... , xn -> an , then Xi zb x2 ± ••• zb xn — a1 i a2 i t . . . zb an. T his theorem is frequently form ulated as follow s: the lim it of an algebraic sum (w ith a constant nu m b er of terms) is equal to the algebraic sum of lim its; this form ulation is even m ore obvious if the theorem is w ritten dow n in its equivalent fo rm : lim (*i

x2

dz

zh

•••

zb *«) =

lim xi

±

lim x2 zb ...

zb

lim # n.

I t is only necessary to rem em ber th a t it is assumed th a t each term has a lim it; this is the necessary req u irem en t o f this th eo rem ; on the other h a n d the existence of a lim it of the algebraic sum as a whole is th en no longer assum ed b u t m ain tain ed (and, of course, proved). T he full (b u t ra th e r lengthy) form ulation of theorem 1 should read as follow s: I f in a certain process every quantity x%(1 ^ i ^ n) has a limit, then the algebraic sum o f these quantities also has a limit and this limit o f the algebraic sum is equal to the algebraic sum o f the limits o f terms This note also refers to all subsequent theorem s of this kind. Proof. I t follows from the assum ptions m ade w ith regard to this theorem th a t in the given process all the differences Xy

flj

—oq, x2

a2 : CC2 , ••• j xn

an

ccn

are infinitesim al quantities. I t follows from theorem 1, § 8 th a t th eir algebraic sum a A ± a2 zb . . . -L a* is also an infinitesim al q u an tity b u t this algebraic sum is, evidently, equal to (*j

zb

x2

zb

± .* n ) “ (ai ± az i

. . . zb

an) ;

from w hich it follows directly th a t *1 ±

X2 ±

. . . zb

Xn

ax zb a2 zb . . . zb aM

a n d the present theorem is thus proved.

34

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS Theorem 2.

I f in a certain process xx -> al9 x2 -> a2i ••• > *n

then xt x2 ... Xn

ai a2 ... an.

Procf. T o begin w ith, let us prove this theorem for two factors (n = 2). L et us assume th a t — flj = «i, * 2 — 02 = a 2 a n d th at oq and a 2 are infinitesim al quantities. H ence X\ = y a n ; we thus have an

^

a n d the sequence (1) becomes 1

.

1

9n }

50

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

W e evidently have an -+ 1 (n —►oc ).

T h u s owing to the fact th a t

\a n - l\ = } p t ( n = 1 , 2 , . . . ) , a n d no m atter how small e > 0, we can choose n0 so g reat th a t 1 / 2U° < £j so th a t for every n ^ n 0 *• j ^ n ~~ 1 |

2. The one-sided limit of a function. W e shall now consider th e second basic type of process, i. e. a process in w hich the basic v ariable x varies continuously, i. e. it runs th ro u g h all interm ediate values; in doing so it can either increase indefinitely or decrease inde­ finitely an d thus rem ain lim ited, or it can becom e an indefinitely large quan tity , i. e. its absolute value can grow indefinitely. E ach case m ust be considered on its ow n m e rits ; how ever, all cases have m any com m on characteristics w hich enable us to tre a t them m ore briefly. In every case the q u a n tity y w hich p articip ates in a process is assumed to be a n a rb itra ry function y — f (*) o f the basic variable x an d we are here trying to explain clearly the exact m ean in g o f the statem ent “ in the given process the q u an tity y tends to the lim it L et us first consider the case w hen the basic v ariab le x grows indefinitely a n d is a positive n u m b e r ( x —►+ oo)*. T his case is very close to the previous case a n d the only difference is due to the fact th a t here * runs th rough all in term ed i­ ate values in the process of its grow th, w hereas in the e th e r exam ple n could assume only integral values. As before the w ords “ from a certain m om ent onw ards” m ean “ beginning with a certain value A of x and for all its greater values” . T h e exact m eaning o f the statem ent lim j; = b co

or

y

b { x —> co )

x

here is as follows : 720 matter how small the positive number z is, there exists a positive number A such that \y — b | < z fo r any x ^ A. In cases w here the basic variable * is a negative infinitely large q u an tity ( x — 0 0 ) , i.e. alth o u g h it is a negative q u an tity its * T h e n o ta tio n * —» c o o r * - » +

00

is u sed to d e n o te th e in d e fin ite g r o w th o f

a;, b o th w h e n this g r o w th takes p la c e b y ju m p s a n d w h e n it is c o n tin u o u s ; for this reason th e a c tu a l t y p e o f v a r ia tio n sh o u ld in e a c h ca se b e in d ic a te d in th e te x t.

A C C U R A T E T H E O R Y OF L IM IT T R A N SIT IO N

51

a b so lu te value grows indefinitely, the relationship y -> b is obviously •defined in the same w a y : the relationship lim y = b

or

y

b (x

— oo )

x —> — CO

1 m eans th a t no matter how small z > 0 may be, there exists a positive number .A such that | y — b | < z f or any x ^ — A. L et us now consider the case w hen the basic variable changes continuously (i.e. increases or decreases continuously) an d rem ains, a t th e sam e tim e, a lim ited quantity. W e shall learn in chapter IV th a t an this event x tends to a certain lim it a. I f x grows indefinitely, th e n it approaches a from the side of the lower values (‘‘from the deft” ) an d this is usually den o ted by : * —►a — 0. I f * decreases •continuously th en it always rem ains g reater th a n the nu m b er a an d -approaches it from the side of the greater values (‘‘from the rig h t” ) ; th is is denoted as : x -> a 4- 0. T o begin w ith let us consider th e first case {x < a, x -> a — 0 ). T h e words “ from a certain m om ent •of the process onw ards” evidently m ean here “ beginning w ith a •certain value a — 8 < a of x, an d for all values closer to a (and, o f •course, sm aller th a n a ) ; m ore briefly we can s a y : “ for all values of -X w hich satisfy the inequalities a — 5 x < a” . T h e exact m ean­ in g of the statem ent lim y = b x «—0

or

y

b (x -> a — 0)

‘is as follow s: no matter how small the number z > 0, there exists a positive number 5 such that \ y — b | < z for any value o f x which satisfies the inequalities a — S ^ x < a . T h e exact m eaning of the statem ent given below is determ ined i n a sim ilar w ay lim

y = b

or

y

b {x

a

0)

a n d is form ulated in exactly the same w ay except th a t the inequality a — 5 < x < a m ust be replaced by a < x < a + 8. W e have thus established the exact m eaning o f the concept of lim it for all processes of the basic types w hich are used in m a th e ­ m a tic a l analysis. L et us em phasize once again the fact th a t all a rg u m en ts a n d results obtained in c h a p te r II w hich w ere the same fo r all types of processes, are, of course, also valid for o u r new an d m ore exact definition of lim it tra n sitio n s; the new definition does n o t

A C O U R S E O F M A T H E M A T IC A L A N A L Y SIS’

52

contradict in any way our old definition w ith w hich it is co m p atib le— it only provides m ore exact specifications for cases of different kinds. L et us now m ake one m ore rem ark. L et us assume th a t theq u an tity .y w hich participates in a certain process, does not tend tow ards a lim it, b u t continues to grow indefinitely. L et us assume* th a t we are dealing w ith a process of the type a* -> a — 0 ; we can,, therefore, w rite y

+ oo ( A' —> a — 0 ).

W h at is the exact m eaning of this statem en t? F ro m all th a t wassaid above, we are able to answ er this question w ithout d ifficu lty : nomatter how small A > 0, there can be found a S > 0 so that we have y > A'l fo r all values o f x which satisfies the inequalities a — 5 ^ .v < a . By using this as an exam ple, the read er will be able to find for him self the exact m eaning of the relationships y + oo an d y —> — oo in any process of the type considered above. H e will find this to b e ­ an excellent exercise.* § 15.

The Development of the Concept of Limit Transitions

T h e two types of lim it transitions (the lim it o f a sequence a n d ’ the one-sided lim it of a function), ;;hich we considered in g re a t detail? in the previous p a ra g rap h , are of basic im portance in m a th e m a tic a l1 analysis, for, all o ther m ore com plicated types of processes w hich weshall encounter in future can be broken dow n to those cases. H ow ­ ever, to m ake this reduction possible in every case, we m ust now develop som ew hat the concept of a q u a n tity w hich tends tow ards a lim it in a given process. L et us begin by considering a simple exam ple w hich w ill show us the necessity and the course of this developm ent. L et us assumeth a t the process in w hich we arc interested involves infinite decrease of the perim eter p ( ‘basic” variable) of a certain rectangle w here the form of this rectangle can change in the course o f the process in any w ay we please. O w ing to the fact th a t in a rectangleof perim eter/?, each side is sm aller th a n p j 2, the area s ol the rectangle of perim eter p will always be sm aller th an p 2 / 4 . W hen p —>- 0, we * Of- p ro b lem s

3 4 9 -3 5 2 , se c tio n 1 o f th e ‘p r o b le m b o o k ’ b y B. P . D e m id o v ie h

m e n tio n e d in th e p refa c e. T h e n u m b e r s o f th e p r o b le m s as t h e y a p p e a r in th e se co n d ed itio n , ca n b e fo u n d o n p . 622."

-ACCURATE THEORY OF LIMIT TRANSITION

53

-evidently have p 2 j 4 —>■0 ; therefore, the area s in o u r process (i.e. w hen p —> is an infinitely small qu an tity , a n d we can write s -> 0 (p

4-0).

T h e exact m e an in g o f this statem ent is determ ined in the usual w ay : . no matter how small z 0, there exists a 5 > 0 such that the area o f any -rectangle, the perimeter o f which is smaller than 6^ will be smaller than z. This exam ple differs basically from all the exam ples w hich we -considered so far. T his difference is due to the fact th a t for a given perim eter p, the a rea s of a rectangle can have an infinite n um ber of -different values, so th a t s is not a function o f p. O w ing to the fact th a t we have taken p as the basic variable in our process, an d also because so far we assum ed th a t the q u a n tity p articip atin g in a given process is a function of the basic variable, we cannot, strictly speaking, con­ sider s to be a q u a n tity w hich participates in our process; it is even less possible to speak of its tendency tow ards a lim it. H ere we are - dealing w ith a q u a n tity whose value a t every m om ent of the process (i.e., for every value of/>) rem ains indefinite. A t the same tim e it is still true to say th a t provided the perim eter p of the rectangle is • chosen sufficiently sm all, the area s of this rectangle, no matter what infinite number o f possible values it may assume, will be as small as we please. M ore e x a c tly : no matter how small z > 0, there is a 8 > 0, such that fo r any rectangle with perimeter p a).

(2)

T h e notation x —>- a, in contrast to the form er notations x —> a — 0 an d x —>■a + 0, shows th a t in this case, * approaches the nu m b er a , b u t th a t it m ust not necessarily increase or decrease : it can change the direction of its tran sitio n and, in particu lar, become g reater or sm aller th an a. T herefore, the lim it o fy w hen x -*■ a, w hich we ju st described, is know n as the two-sided limit o f a function. Let us rem em ber once again th a t the exact m eaning of the lim it transition (2 ) involves the following : no matter how small e > 0, there is a 8 > 0, such that fo r any value o f x fo r which 0 < | x — a j ^ S, we have | y — b | < e. L et us now m ake one m ore im p o rtan t rem ark : in order that the number b should be the two-sided limit o f y when x —»- a, it is necessary and sufficient that the one-sided limits o f y 3 viz. Hm y and Urn y , should exist x -> a- \- 0

and be equal to b. If

x —> a — 0

In fact let us assume th a t s > 0 is given arb itrarily . l i m y = b, x

a

th en provided we have a sufficiently small 8 > 0, it follows from 0 < | x — a | ^ 5 th a t \ y — b | < s. But if a < x ^ a + 5, then even m ore so, \ x — a \ ^ S, and therefore, we also have | y — b | < e. T h u s whenjy —>■ b (* a + 0), it can be shown sim ilarly th a t this is also true w henjy -> b (x a — 0). L et us now assume, conversely, th a t we are given y -> b> when x -> a T 0, a n d w hen x a — 0. In th a t case, no matter how small s > 0 we can find a 8 X which is such that | y — b | < e, when a < x ^ a + 8j, and such a 82, that \ y — b \ < e when a — 8 2 < ^ ; k < 0 ; if we denote by 8 the sm aller of the num bers 8 j and 8 2, then a — a n d we have | y — b [ < t ; this shows th a t y —> b (.x —> a), w hich h ad to be proved. W e can thus see th a t the process o f two-sided ap p ro ach o f the v ariable a: to the lim it a simply involves the process of the one-sided approaches x a + 0 and x —> a — 0. T his is the first exam ple illustrating w hat was said in the note in § 14, i.e. th a t different types o f analyses can be reduced to the study of the two basic types of processes.

C H A P T E R IV REAL NUM BERS § 16.

Necessity of Producing a General Theory of Real Numbers

O ne characteristic of a variable q u an tity arises from the fact th a t in the course of a process it assumes different values. E ach o f these values is expressed in term s of some num ber. If, for exam ple, tem perature of a ir rises from 5 to 10°C, we n atu rally assume th a t in the course of this process it runs gradually th rough all the num bers from 5 to 10. But w hat exactly do we m ean by “ all the n u m b ers” ? It is evident th a t these num bers are not restricted to integers alone, for obviously there are m om ents w hen the tem p eratu re is equal to 6*5 °C . Do we, perhaps, m ean “ all integers an d all fractions” ? T h e set o f all integral an d fractional num bers (positive, negative a n d zero) forms the so-called set o f rational numbers; these num bers and the operations which can be perform ed w ith them are studied in detail, in arith m etic and algebra. T h e question w h eth er these num bers are sufficient for m easuring all the quantities w hich we are likely to m eet in the study of the w orld aro u n d us is o f g reat im por­ tance, both in m athem atics as well as in the accu rate study o f n a tu re . In ancient Greece (probably in the Pythagoras school) a rem arkable discovery was m ade, viz• th a t certain simple geom etrical constructions lead easily and irrevocably to quantities w hich can n o t be m easured w ith the help o f rational num bers. A simple case o f this type is very w ell-know n: if each side adjacent to the rig h t angle in a right-angled triangle is of u n it length, then according to the theorem of P ythagoras, the hypotenuse of this triangle should be such th a t its square is equal to 2. B ut it is easy to show th a t there is no ratio n al n u m b er whose sq u are 56

R EAL N U M B E R S

57

is equal to 2 *. T herefore, if we w ant to restrict ourselves to rational num bers, we m ust ad m it th a t the hypotenuse of the triangle in ques­ tion has no le n g th ; obviously we cannot arrive a t this conclusion, for geom etry cannot be based on this absurdity. Circum stances in the outside w orld thus m ake it impossible for us to restrict ourselves to the set of ratio n al num bers alo n e; it is therefore necessary to ad d a new type of num bers w hich we shall call irrational. O ne such num ber is ■\/2) the square of w hich, by definition, is equal to 2. However, it m ust be rem em bered th a t the introduction of a new n u m b er is an easy m a tte r w hich does not by itself have any significance; if we wish to m ake this new ly-introduced num ber a fully valid n u m b er of the fam ily of num bers, we m ust, to begin w ith, define its position in this fam ily, i.e. we m ust determ ine w hich rational num bers are sm aller a n d which %g reater th a n -y/2. In the second place, we m ust define all operations to w hich this new num ber can be subjected, (for we do not know, for exam ple, w hat is m eant by •\ /2 + 1, 3 \/2 , 1 / a / 2 ) , a n d we m ust prove th a t these operations are subject to the same laws w hich govern operations w ith rational num bers (for exam ple, we m ust show th a t \ / 2 1 = 1 + \/2 ) . All this can be done, though it w ould necessitate considerable e ffo rt; how ever, the object we have in m ind fully justifies this procedure. But let us assum e th a t we already did all this. Sooner or later we shall m eet an o th er physical or geom etrical problem w hich will make necessary to introduce an o th er new n um ber, the square of which is eq u al to 3 or 5, etc. It would thus no longer be possible to repeat in each case the same chain of argum ents w hich we used for m aking ■y/2 a fully valid num ber- L et us now assume th a t we have found a w ay w hich w ould enable us to use a single m ethod for introducing square roots of all n a tu ra l num bers into the fam ily of num bers (this is, no d o ubt, possible). T h e possibilities of applications are hereby n o t exhausted. If we are trying to find the length of the side of a cube, whose volum e is equal to 2 m 3, we m ust introduce the n um ber ^ /2 . A nd even if we do introduce roots of any degree of any ra tio n a l n u m b er into the fam ily o f num bers, this will not be sufficient. O n one h a n d , the required n u m b er is frequently defined as a root of a given e q u atio n : on the o th er h an d , we know practically *

q = 2 sq

If

we

h a v e {p j q) 2 = 2 ,

th e n w e find that p 2 = 2 q 2 ; let p — 2 Tp ' ,

w h e r e p ' a n d q ' a re n o lo n g e r e v e n n u m b e r s.

p 2 = 2 * r p ' z,

T h en w e have

2 ? 2 — 2 2 H 1 q'2f

a n d th e e q u a lity p 2 = 2 q 2 le a d s to a c o n tr a d ic tio n , for its left sid e co n ta in s 2 w ith a n ev e n in d e x a n d th e rig h t sid e c o n ta in s 2 w ith an o d d in d ex .

58

A C O U R S E O F M A T H E M A T IC A L A N A L Y SIS

th a t there m ust be one such root, whereas theory shows th a t there is no such root am ong all possible ratio n al an d irratio n al n u m b e rs ; an d again we find it necessary to introduce a new n u m b er which we simply define as the root of our equation. H ere again we m ust repeat the same argum ent w hich we used above in connection w ith the num ber \/2 . In practice, even the simplest geom etrical problem s m ay lead to difficulties o f this kind. This is p articu larly shown by the following exam ple in w hich we try to find the area of a circle of unit radius. We know th a t the area of a circle is defined as the lim it of the areas o f all inscribed (or circum scribed) reg u lar polygons w hen the num ber of sides of these polygons increases indefinitely. W e know from this visual representatio n an d practice th a t a circle has an a r e a ; in reality, can we reconcile ourselves to the fact th a t such a simple figure as a circle has no area a t all ? At the same tim e m athem atics tells us th a t there is no such lim it am ong all the num bers so far handled, including the roots of all algebraic equations. H ence we have no alternative b u t to introduce a com pletely new num ber for m easuring the area of o u r circle an d rep eat once again the chain of argum ents m entioned above, in o rd er to m ake this new num ber a fully valid m em ber of our extended fam ily of num bers. This new num ber is no other th a n the w ell-know n n u m b er tt. The above examples clearly show th a t the p rocedure is u n ­ scientific and unpracticable, if, in order to solve a problem , for whose solution the existing num bers are insufficient, we find it necessary to introduce new num bers, define th eir position am ong the existing num bers, find and investigate the operations w hich can be perform ed w ith them , etc., in other w ords— to do all th a t is necessary to m ake them fully valid m em bers of the fam ily o f num bers. I t is thus quite clear th a t a general theory o f irrational numbers m ust be p ro d u c e d ; we m ust find one general principle o f origin of irratio n al num bers (the num bers studied so far are p a rtic u la r cases) w hich w ould include all the historically known exam ples of this kind, an d thus g u aran tee th a t it will no longer be necessary to introduce fu rth er new irratio n al num bers. F or num bers originated by this general principle, it will be necessary to repeat all argum ents in general form, b u t this will in future enable us to operate With them in the sam e w ay as we do w ith ratio n al num bers in elem entary arithm etic an d algebra. T h is is the only scientific ap p ro ach to the problem in question. All this w ork is not a p a rt o f m ath em atical analysis— a science w hich deals w ith changes in qu an tities—b u t is p a rt o f the theory of

REAL NUM BERS

59

n u m b e rs; how ever, until this problem is solved, m athem atical a n a ­ lysis can have no stable b a sis ; in fact, as we have already said a t the b eginning of this p a ra g rap h , the values of all variable quantities are expressed in term s o f n u m b e rs ; therefore, we cannot even begin the study of variable quantities w ithout know ing the num bers w hich m odern m athem atics has a t its disposal a n d the properties of this set of num bers. A short outline of the m odern theory o f this set of num bers is given in the next few p arag rap h s of this chapter. § 17.

Construction of a Continuum

1. W hen we evaluate \ / 2 w ith the help of conventional m ethods we obtain the following sequence o f approxim ations for this n u m b er : a0 — 1;

a x — 1*4;

a 2 = 1*41 ;

a 5 = 1*414; . . .

E ach one of these values is a ratio n al n u m b er (a finite decim al fraction) an d each n u m b e r is g re a ter th a n the preceding n u m b er (or, a t le a s t,. is equal to it). T h e squares of these num bers tend to 2. * a2n —> 2 (n -> oo ). H ow ever, the num bers a n cannot tend to a rational lim it: if such a lim it r exists, then a n -> r w ould im ply a n 2 —> r 2, an d since a n %-> 2, we w ould have r 2 = 2 ; b u t this w ould m ean th a t a ratio n al n u m b er r exists such th a t its square is equal to 2, w hich, as we know, is incorrect. We thus have the sequence a o* a i, a 2 , • • . 3 o, n, • • •

(1)

of w ell-know n n u m b e rs; this is an increasing sequence, i.e. we always^ have a n + x ^ 2 ; h e n c e

a2n < 2 < (a n +

,

a n d th e r e fo r e , 2

0 < 2 — a2n
00 ) .

*60

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

\ / 2 whose square is by definition equal to 2, we are filling in, as it w ere, a gap existing betw een rational n u m b ers: o u r n u m b er m ust fill ?this gap in the set of rational num bers and be defined as lim it o f the increasing sequence (1). T h e situation created by the in tro d u ctio n of the irratio n al fnum ber tt is very sim ilar to the one w hich we have ju st described. L et us assume th a t the area of a regular /z-gon inscribed in a circle •of unit radius is equal to s n ; in this case the num bers •••

(2)

form an increasing sequence an d the n u m b e r tt is defined geom etri• cally as lim it o f this sequence. H ere the position is som ew hat com pli­ c ated by the fact th a t the area s n is, generally speaking, expressed in terms of irrational n u m b e rs; how ever, these num bers are am ong the simple irrational num bers an d can easily be expressed in term s of roots of n atu ral n u m b e rs; we can, therefore, assum e th a t the a re a sn is expressed by a w ell-know n num ber. I t now ap p ears th a t the •sequence (2) has no lim it either am ong rational num bers or even am ong the num bers of the w ider class in term s of w hich the area s n is expressed. T hus by re-introducing our new n u m b er zrwe are filling, as it were, a gap existing in the set of all the num bers we have m et so far, and this num ber is the lim it of the increasing sequence (2), i.e. it is a lim it w hich d id /not. exist am ong the num bers we have 1known so far. Let us assume now th a t we are given an a rb itra ry increasing sequence f 1} 1 2’ • • • >

? • • • 0 n + 1 ^ ^ m)

(3)

o f rational num bers. T o begin w ith, we m ust distinguish two cases : the n u m b er r n can grow indefinitely as n increases; or a positive n um ber C can exist such th a t r n < C for any n. In the first case r n is an infinitely large quantity w hen n -> oo, an d therefore, it can n o t tend to a limit. W e shall, therefore, concentrate on the second case, rem em bering, th a t for the m om ent we only have ratio n al num bers a t our disposal. In the case u n d er consideration the sequence (3) is -bounded; it m ay happen, however, th a t it has a ratio n al lim it r ; thus the sequence

REA L NUM BERS

6L

is a n increasing bounded sequence w hich tends to unity as its lim it: r n = 1 ----- — -»■ 1 n

(n v

oo ).

I t m ay also h ap p en th a t the bound ed increasing sequence has • no ra tio n a l lim it; thus, for exam ple, the sequence (1) o f ap p ro x im a­ ted values of \ / 2 is evidently an increasing b ounded sequence (all' a n < 2), b u t a t the sam e tim e we have seen th a t it has no lim it. L et us now agree (in the sam e w ay as we did w hen introducing the irratio n al n u m b er \/2 ) th a t every time when we deal with a bounded' sequence (5) o f rational numbers, fo r which there is no rational limit, we shall' take a new irrational number as its lim it* W e have thus established a general principle o f origin of irrational num bers. H aving m ad e th is . agreem ent, we have also defined the w hole set o f irrational num bers. W e shall see later th a t the set satisfying this definition has, in fact, taken some final f o r m ; in future we shall not introduce other new num bers a p a rt from those defined by o u r agreem ent. 2. Example. L et us assum e th a t a n = (1 + 1ln)n, {n = 1, so th a t all the num bers a n are ratio n al (a1= 2, a 2 = 9/4, a 3 — 64/27, etc.). W e will show th a t the sequence of num bers a n is an increasing b o unded sequence and th a t has an u p p er lim it. A ccording to the bin om ial form ula we have : n

* W e sh a ll sh o w at th e e n d o f th is p a r a g r a p h th a t th is n u m b e r d o es, in f a c t ,, sa tisfy th e d e fin itio n o f a lim it.

62

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

Sim ilarly,

A com parison of the rig h t-h an d sides of the form ulae (4) an d (5) shows th a t in the sum (5) each term is g reater th an its corresponding term in the sum in form ula (4), since the replacem ent o f n by n + 1 causes a n increase in each sm all brack et in form ula (4 ); m oreover, there is an additional term in form ula (5) corresponding to k = n -+- 1 w hich is absent in form ula (4). T herefore d n+

1

@n

( ^

1

s " ? • ■ •) 3

.i.e. th e sequence of num bers a n is increasing. form ula (4) th a t for any n

I t also follows from

n

k= an d owing to the fact th a t k \ ^ n

from a certain n u m b er k onw ards, are equal to one another, i.e. r* = r* + 1 = rjt +z = then evidently a = ry an d a is a ra tio n a l num ber. T his •sequence r n is called stationary sequence; it is obvious th a t if a is an irratio n al n um ber, the sequence determ ining this n u m b e r is never •stationary; how ever, if a is ratio n al, the sequence r n can be statio n ­ ary [ r n = a, n = 1 , 2 , . . .], or non -statio n ary [rn = a — 1 /« , ,n = 1, 2, . . .]. T his shows th a t in the construction o f continuum , we could restrict ourselves to the consideration of n o n-stationary, increasing sequences o f ratio n al num bers. L et us assume th a t we are given two non-stationary increasing Abounded sequences of ra tio n a l num bers ^1 j b ) • • • ) ? n i • • • j s 1 t s 2 > • *

• j $n j • • •

W is)

64

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

As we know, each sequence originates a real n u m b er w hich can rational or irrational. L et us assume th a t a is this n u m b er [for sequence (r)] an d (3 [for the sequence (.y)]. W e m ust now solve problem , w hich of the three possible relations a < p , a > p , a = applies in this case.

be th e th e p -

L et us agree to say th a t the sequence (j) is a major (exceeding) sequence as com pared to the sequence ( r ) , if for every n u m b e r r n o f the sequence (r), a num ber s m of the sequence fy) can be found such th a t s m ^ r n (m eaning o f this inequality is clear, since the num bers r n and s m are rational). W e can have four different cases : ( 1 ) (,y) is m ajor as com pared to (r) an d (r) is m ajo r as com ­ pared to (.y); (2 ) (s) is m ajor as com pared to (r) b u t (r) is not m ajo r as com pared to (j'); (3) (r) is m ajor as com pared to (s) com pared to (r);

b u t (s) is not m ajo r as

(4) (,y) is not m ajo r as com pared to (r) an d (r) is not m ajor as com pared to (.y). It can readily be seen th a t the fo u rth case is im possible. In fact, if fy) is not m ajor as com pared to (r), then a n u m b er r n can be found such th a t s m < r n for any m; b u t it is obvious th a t in this case (r) is m ajor as com pared to (s). W e, therefore, only have to consider the first three cases. In case (1) we are assum ing th a t a = (3, in case(2) th a t a < ,3 and in case (3) th a t a > (3. These assum ptions u n i­ quely define w hich of the th ree relations applies to any p a ir o f real, num bers. It can readily be shown th a t in the event w hen b o th num bers a and (3 are ratio n al a n d (r) an d (s) are strictly increasing sequences, the above concepts o f equality an d inequality w ould, as-.expected, coincide w ith the conventional concepts. W e m ust now find out w hether the definitions o f eq u ality a n d ’, inequality o f real num bers, as given above, possess the same p ro p er­ ties as those for ratio n al num bers. L et us consider, for exam ple, th e transitive property, w hich is d u e to the fact th a t a ^ p an d p ^ y implies a ^ J . T his is a well-known p ro p erty o f ra tio n a l numbers;:, b u t for real num bers it m ust be proved on the basis o f o u r definition o f equality an d inequality. T his can be done qu ite easily : to begim w ith, we m ust establish the transitive p ro p erty o f m ajority, i.e. if' (s) is m ajor as com pared to (r) an d {t) is m ajor as com pared to (.y^ then (t) is m ajor-as com pared to (r).

REAL NUM BERS

65

4.

A fter establishing all necessary properties o f equality a n d in eq u ality , the theory of continuum tries to establish the operations to be perform ed w ith real num bers, e.g. how to determ ine the sum a -f- P o f two real num bers. Let us assume th a t a is defined by the sequence (r) an d (3 by the sequence (s); then r 1 4" Si y r 2 +

s 2 3 • • • j rn +

s n, • • • »

(0

is evidently an increasing bounded sequence o f ratio n al n u m b e rs; the real n u m b er y thus defined is natu rally the sum a + [3 of the num bers 0 an d x is any real num ber (the d e fin itio n . of an exponential fu nction). L et us assum e th a t a > 1 ; then the ratio n al a x is also defined for any .v, an d is a ratio n al function o f x; in fact, if r = p ! q and r ' — p ' / q are rational num bers and if r < r ‘ 1

1

th en a q > 1 an d therefore a r =

(l ^

p q^

1

p/

= ar

L et us now assume th a t the real n u m b er a is defined by the increasing bounded sequence (r) of ra tio n a l num bers. In this case th e sequence

a \ a r 'z ,

. . . ,

a ln, . . . 9

h evidently bounded, and it follows from the above p ro o f th a t it is also an increasing sequence. H ence it defines some real nu m b er w hich is evidently the n u m b e r aa. In this w ay th e exponential function ax is defined for any real v; at the sam e tim e we also estab­ lish the fact th a t this is an increasing function (if a > 1 ), an d a de­ creasing function (if a < 1). A logarithm ic function is defined in sl sim ilar way.

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

66

It follows from these definitions th a t the know n properties o f the rational values of argum ents of these functions also apply to a ll real values; thus we have in all cases a x +v — a x a* , lo g a (xy) = loga * + lo g a j, etc. W e are unable to pay greater attention to these problems, w ithin the scope of this course. 5. T h ere is, however, one m ore point w hich m u st be clarified. In the above w ording o f the principle o f origin o f real num bers, we said th a t the n u m b er a has originated from the increasing bou n d ed sequence of rational num bers 'i , H , . . . , rn , . . .

(O'

an d is assumed to be limit of this sequence. T o convert this state­ m e n t into a real tool o f investigation, we m ust prove i t ; having le arn t th e arithm etic of real num bers, we are now, in principle, able’ to d o ­ it. I t is evidently necessary to prove th a t no m a tte r how sm all e > 0, we shall have for all sufficiently large n : a — r n < s. T o begin w ith, let us prove the following auxiliary result onsequences o f rational num bers. Lemma. Let (r) be an increasing bounded sequence o f rational numbers. In that case, no matter how small s > 0, there is an index n 0 , such that 7i > n 0 for m > n 0, and we always have r n — r m < s. Proof. L et us assume th a t the statem ent expressed by the lem m a is incorrect, i. e. th a t there is an £ > 0 such th a t the in­ equality Ym ^ ^ is satisfied for all values of n a n d m w hich can be as large as w e please. In this case, no m a tte r how larg e th e n a tu ra l n u m b er k, there are k pairs o f indices (wz* , n j ( 1 < i < k) such th a t m 1 < n 1 < m 2 < n 2 < - . • < mi- < n

and Tn . *

Tm ^ £ i

(1

^ i ^ k);

b u t in this case rn

k

— r m 1 =^ \( r n k \

k/

— t„

\

k

~

Ta

k —?

) + (\ r rck —1

/

~~ r m

\

k —\ )

\ + • • • + /^ r**n j — rTm \^ > k e.

REAL NUM BERS

67

for the first, th ird , fifth, etc. brackets are not sm aller than z a n d the second, fourth, etc. brackets are positive. H ence kz

r m

O w ing to the fact th a t k can be as large as we please, the sequence (?) will contain term s w hich are as large as we please, an d this con­ tradicts the fact th a t it is a bounded sequence. T his proves our theorem . L et us now assum e th a t z is an a rb itra ry rational n u m b e r; it follows from the above lem m a th a t r n — r* < e for a sufficiently large k a n d for any n ; therefore, the sequence

is a m ajor sequence as com pared w ith r i — rk , r 2 — rk , . . . , rn — rk , . . . , w hich evidently gives the real num ber a — r k ; owing to the fact th a t the sequence (e) gives the n u m b er s, we have by virtue of the definition of inequality of real num bers a — rk < z . for sufficiently large k. This proves the proposition for a rational z ; b u t owing to the fact th a t for any real z > 0 , we can find e ' > 0 , sm aller th an e , we w ould prove the proposition for any real £ > 0 . W e m ust also d raw atten tio n to the fact th a t the principle of origin of irratio n al num bers accepted by us is by no m eans the only possible m ethod ; in the second h a lf o f the last century, when the necessity o f p roducing a general theory o f real num bers becam e a p p aren t, several such theories were advanced alm ost sim ultaneously a n d each theory h a d its own principle o f origin ; it later becam e obvious th a t all these theories are basically equivalent so th a t the choice of theory should be governed no t so m uch by principle as by the convenience o f the m ethod o f tre a tm e n t an d its applications.

6.

T h e w ider set o f num bers w hich we are now studying, is, as we know , by no m eans the first in the historical developm ent o f nu m bers. T o begin w ith, we learn ab o u t the n a tu ra l num bers in a rith m etic, to w hich subsequently zero, negative an d fractional n u m bers w ere added. T hus as a result o f successive additions, the set of ratio n al num bers is obtained. O u r principle o f origin adds to

68

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

them all irratio n al num bers a n d thus develops it into the set o f real num bers, i.e. into the continuum . W e know th a t all previous additions to the set o f num bers were prom pted, to g re a ter or lesser degree, by o u r wish to be able to perform some operations u n d e r all possible conditions, w hich could otherwise no t always be achieved w ith the help of the older system of num bers. T h u s the in tro d u ctio n o f zero an d negative num bers enabled us to deal w ith all cases o f subtraction ; the introduction of fractions produced the same result w ith reg ard to division (with the exception of division b y zero w hich, by the w ay, still rem ains impossible even w ithin ou r new system o f real n u m b e rs); the introduction o f irrational num bers was p ro m p ted by o ur desire to be able to extract roots. This tendency to deal w ith operations, w hich could not always be perform ed otherw ise w ithin the existing set of num bers, was prom pted in m athem atics not so m uch by an ab stract arg u m en t leading to a form al goal (as it is som etim es believ­ ed), b u t by p ractical necessity ; this is best seen by exam ples like those introduced a t the beginning o f this c h ap te r ; thus we w ere unable to obtain results in cases w here the len g th of the diagonal o f a square of u n it side, was required, or, w hen we w ere trying to find the area of a circle of u n it radius, because the set of num bers a t o u r disposal was insufficient for this purpose. A strict scrutiny of our principle of origin thus shows th a t the introduction of the w ider set of num bers was p ro m p ted by o u r wish to be able to perform certain operations u n d e r all circum stances whereas this could not always be achieved w ith the help o f ratio n al num bers alone. This involves the creation o f a li?nit o f a bounded increas­ ing sequence o f numbers. T his is no longer a n a rith m e tic al operation. O n e of the characteristics of an a rith m e tic al operation is th a t all arithm etical operations are always perform ed w ith a finite group o f num bers ; on the other h an d , our operation requires the existence o f aninfinitesequence of num bers, an d w ith the help o f all these num bers a new n u m b er is originated, w hich is the lim it o f this sequence. T his is an analytical operation, i.e. one of the first an d sim plest operations o f m athem atical analysis. T he w idening of the existing set o f num bers w hich was u n dertaken in order to g u a ra n tee perform ance o f an y o p eration w ith num bers achieves its goal only if the op eratio n is possible w ithin the w ider range of num bers. W e m ust therefore convince ourselves th a t every bounded increasing sequence has a limit within the range o f real numbers. H ow ever, this can readily be proved. In fact let *1 >

>• • • j

n ) • •

( 6)

REAL NUM BERS

69

be one such sequence, i. e. a n + j ) a „ for any n, an d there is a n u m b e r C w hich is such th a t a n < C for any n ; here all v.n are a rb itra ry real num bers. I f from a certain place onw ards, all the num bers in the sequence ( 6 ) are equal to one another, th en the com m on value of these num bers will evidently be the lim it of the sequence (6 ) ; therefore, we can right from the beginning reject this case and assume th a t the sequence (6 ) contains an infinite n u m b er of different unequal num bers. L et us assum e th a t these different u n eq u al num bers in the sequence (6 ) increase in the following order P1

5P

2 • • • • >

j • • • (P n + 1 >

P «)•

D enoting by r n any ratio n al num b er betw een (3 n an d [3n + 1 *, w e have /

Pi
0. Sim ilarly we say th a t the n u m b er a is th e lower bound of the set M if this set contains no num bers sm aller th a n a, b u t there exists a m em ber of the set w hich is sm aller th a n a -{- e for sm all e > 0. It is thus obvious [that the u p p er b o u n d is the sm allest n u m b er w hich do n ot exceed any n u m b er o f the set M ; a sim ilar case applies to the lower bound. Exam ple. T h e set of positive ratio n al num bers whose squares are sm aller th an 2 has 0 as its lower b ound and \ / 2 as its upper bound. In general, b o th u p p er an d lower bounds of th e set M m ay o r m ay n o t belong to this set. T h e upper an d lower bounds of a section evidently coincide w ith its ends and alw ays belong to i t ; on the o th er h a n d in the above exam ple the set u n d er consideration does not contain its low er bound (since it is not positive), nor its u p p er b o u n d (since it is not ratio n al). A set w hich has no upper lim it can n o t have an up p er bou n d , for there is no num ber (5 in com parison to w hich all num bers o f the given set are sm aller. For the purpose of analysis it is im p o rtan t to note th a t a set w ith an u p p er lim it alw ays has an u p p e r bou n d (and only o n e ); sim ilarly, a set w ith a low er lim it alw ays have a single low er bound. T he theorem on existence of bounds for b o u n d ed sets(which is by no m eans self-evident) is one of the m ost im p o rtan t properties of continuum . It can readily be shown th a t, for exam ple, the set in ou r last exam ple has an u p p e r lim it, b u t th a t w ith in the region of ratio n al num bers it has no u p p e r bound.

REAL NUM BERS

73:

T his theorem is proved in the same way for the u p p er and' low er bounds so th a t we only need to prove one o f these cases. Lemma 3 (on existence of bounds for bounded sets). M which has an upper limit has a single upper bound.

The set

Proof. L et us say th a t a section is normal if it contains a t least one point of the set M , an d to the right o f this point there is no point of this set. It can readily be shown th a t from th e two halves of a n orm al section a t least one h a lf will always be n o rm a l; in fact, if th e rig h t h a lf contains a t least one point of the set M , then this right h a lf will evidently be a norm al section ; if, on th e o th er h an d , the right h a lf contains no points of the set M , then the left h a lf will be the norm al half. L et us assum e th a t a is an a rb itra ry point of the set M y an d b is an a rb itra ry n u m b er w hich exceeds all num bers of the set M . T h e section {a, b) — A i is evidently n o rm a l; let A 2 denote its n o rm al half, A 3 the norm al h a lf o f the section A«, an d generally, A«+i the n o rm al h a lf of the section A « (« = 1, 2 . . .) . T h e sections A j, A 2j • • • 3 A n . . • form a contracting sequence, an d therefore, according to lem m a 1, they have a single point 3 in com m on. W e now m a in ta in th a t (3 is the u p p e r b o u n d o f the set M . T o begin w ith, we m ust prove th a t there are no points of the set M to the right o f (3 ; let us assume th a t a > [3 belongs to the set M ; each section A n contains the p oint [3; b u t if this is so, it m ust also contain 1 a, for if it were to end m ore to the left, then the point a o f the set M w ould lie to its right, an d it w ould no longer be norm al. T herefore, each of the sections A n contains both points (3 an d a, and therefore its length is not less th a n a —(3; however, this is impossible,. since A n 0 {n -> oo). H ence there are no points o f the set M tO' th e rig h t o f the point (3. L et us now assum e th a t s is an arb itra ry positive n u m b e r; w hen n is sufficiently large, A « < e ; a n d since A » contains (3, all points o f the section A n lie to the right of (3 — s ; b u t since the section A n is norm al, it does not contain a single p oint o f the set M to the rig h t of (3 — e. Therefore, no m a tte r how sm all s > 0 there is a p o int belonging to the set M w hich lies to the rig h t of (3 — e ; this m eans th a t (3 has also the second p roperty o f the u p p e r b o u n d an d therefore it is, in fact, the u p p e r b o u n d o f the set M ; existence o f theu p p e r b o und is thus proved. T h e fact th a t it is im possible for a given set M to have two different u p p er bounds is alm ost self-evid e n t ; i f there were two such bounds [31 an d Pa (Pi < p2)> then it would!

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

'74

follow from the first p ro p erty of the bou n d (3i th a t no n u m b er o f the set M can lie betw een (3 x and p2, whereas, according to the second property of the bound (32, there m ust exist such num bers w hich leads to the required contradiction. L em m a 3 is thus proved. § 19.

Final Points in Connection with the Theory of Limits

In chapter I I we constructed the basic theory o f limits. H o w ­ ever, some of the m ore im p o rtan t propositions of this theory could only be established on a m ore accurate basis w hich we now have at our disposal after having studied continuum and its fu n d am en tal properties. In this section we shall, therefore in a way, supplem ent our present knowledge of limits. 1. T o begin w ith, let us consider changes in the increasing bounded quantities w ithin a w ider scope. If an belongs to an increas­ ing sequence o f real num bers bounded from above, th en a lim an n

oo

m ust exist; (this follows from the last theorem in § 17). But we know th a t a sequence of num bers is the only way to describe a m ath e­ m atical process. I f we are given an a rb itra ry process described by any m ethod, we shall naturally say th a t the q u an tity x w hich p a rti­ cipates in this process is an increasing quantity if for any two given m om ents of the process its value a t the later m om ent is not less th an its value a t the earlier m om ent. W e say th a t the q u an tity x has an upper limit in the given process if there is a num ber C, such th a t from a certain m om ent of our process onw ards we always have x < C. I t is evident th a t the increasing sequence w ith an u p p er lim it, w hich we have considered a t the end of § 17, is a p articu lar case in the general system of increasing quantities w ith u p p er limits. W e shall see th a t the theorem a t the end of § 17, w hich was proved for this p articu lar case, rem ains valid for our general system. Theorem 1. has a limit.

Every increasing quantity which is bounded from above

Proof. O w ing to the fact th a t the q u an tity x is increasing, an d bounded from above, there m ust be a nu m b er C such th a t always x < C; therefore, the set M of the values taken by x is bounded from above, an d , in accordance w ith lem m a 3 § 18, it has an u p p er bound j3. Let £ be a positive num ber w hich can be as sm all as we please. In accordance w ith the second p roperty o f an u p p er b o u n d , there m ust be a num ber in the set M {i.e. x will sooner or later take this value) w hich is g reater th an (3 — £; since x is a n increasing q u an tity ,

H EA L NUM BERS

75

all its subsequent values will be greater Lth an p — e. But it follows from the first property of an u p p er b o u n d th a t no num ber of th e set M exceeds (3. H ence from a certain m om ent onw ard we alw ay have : P — E < X < (3, an d therefore I * — P 1 < s; b u t since the n u m b e r z is as small as we please, therefore in this process .v — p. T his proves theorem 1. I t is evident th a t this theorem rem ains valid for a decreasing q u a n tity w hich is bounded from below. W e have so far said th a t a: increases in the given process if its value * 2 a t a la te r m om ent of the process is never sm aller th an its value .V! a t an earlier m o m e n t: x 2 ^ *i- T hus an increasing q u a n tity either increases in the course o f the process or m aintains its d o rm er value b u t it never decreases; therefore, we can n a tu ra lly say th a t this q u an tity is non-decreasing an d reserve the term “ increasing” for quantities for w hich we always have x 2 > * 1 w ith no possibility o f equality. We shall use this term inology in future. T hus, for exam ple, as .v increases, the qu an tity A x3 also increases, b u t |* | (c.j\ § 4, exam ple 1) is a m erely non-decreasing q u an tity . I t is obvious th a t every increasing q u a n tity is a t the sam e tim e also a non-decreasing q u a n tity , b u t the converse is n ot tru e. Sim ilarly we say th a t x is a decreasing q u an tity if we always have x 2 < .vx an d th a t it is a nonincreasing q u a n tity if we always have x 2 ^ x All non-decreasing a n d all non-increasing quantities are called monotonic (they always change in th e same direction). H ence, in general, theorem 1 can be stated as follow s: a monotonic quantity which is bounded in the direction o f its change always has a limit.

2. L et us now consider a new problem . W e have ju st shown th a t for a m onotonically changing qu an tity , boundedness in the a p ­ p ro p riate direction serves as a sufficient condition for existence o f a lim it. In g eneral, w hen a q u a n tity does n o t change m onotonically, it is often im p o rtan t to find if this q u a n tity has a lim it in the given process. A necessary an d sufficient condition also exists for the general case, as we shall la te r see an d is o f g reat theoretical im por-. tance. W e form ulate an d prove this condition for the general case .as follows :

76

A C O U R S E O F M A T H E M A T IC A L A N A L Y SIS

Theorem 2 (criterion for existence o f a limit). In order that x should in the given process tend to a limit, it is necessary and sufficient that, no matter how small the positive number z,jrom a certain moment o f the process onwards, two arbitrary values o f x should differ from one another by not less than s. Proof. stages.

W e shall break up the condition o f sufficiency into three

1. According to the conditions of the theorem , there will be a m om ent in our process after w hich two values o f .v will differ from one an o th er by less th an unity. I f a t the m om ent in question x — .v0, then for all subsequent m om ents X

Q —

1

X

in w hich each section represents the norm al h a lf of the preceding section. T h e com m on point of all these sections (w hich exists, as . shown by lem m a 1, § 18) is denoted by a. 3. L et us prove, Anally, th a t lim x — a . L et e be an arb itra ry positive num ber. L et n be so large th a t A n < \ e- L et us Ax a m om ent of our process so th a t from th a t m om ent onw ards two a rb itra ry values of a differ from one an o th er by n o t less th a n \ £. Since the section A n is norm al, it contains a value x x w hich x takes after the m om ent in question. H ence for any value x 2 taken after th a t m om ent, we have, as a result of the choice o f th a t m om ent, *1 I < 2£-

R EA L NUM BERS

77

But on the other h an d , since both a an d x x belong to the section A „, w hose length is not less than Js we have I

~ ai
n 0, m > n 0. In other words,

* T h is c o n d itio n is o fte n k n o w n as Cauchy's criterion ; in g en er a l it is u su al to use

th e

ter m criterion in c o n n e c tio n w ith co n d itio n s w h ic h a re sim u lta n e o u sly

m ecessa rv a n d su ffic ien t.

'

78

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

any two “sufficiently far rem oved” term s o f the sequence should differ from one another as little as possible. 2. In order that the function y = f { x ) should have a limit fo r x -+ a , it is necessary and sufficient that the following condition is satisfied : no matter what the positive number e be, there exists another positive number 8 such that we always have \ f ( x x) — f ( x 2) | < e fo r | — a \ < 8 1at2 — a | < 8 (atj a, x 2 ^ a). In other words, the values o f the function f ( x ) a t .tw o different points sufficiently close to a should differ from onean o th er as little as possible. 3. In order that the function y = f (x) should have a limit as x increases indefinitely (* -> T o o ), it is necessary and sufficient that the following condition is satisfied : no matter what the positive number e be, thereexists another positive number A scuh that we always have]f (x j) — f ( x 2) | < £ for Xx > A, x 2 > A. In other words, the values of the function f ( x ) for two sufficiently large values o f x should differ from one an o th er as little as possible. Finally, a t the end of the section on lim its, we find it necessary to say th a t in order to acquire practice in the evaluation o f limits, it is necessary to solve m any examples. M any instructive exam ples of this type can be found in the Problem Book by B.P. D em idovich, in. ■which problem Nos. 38, 40, 41, 42, 48, 50, 50-58, 60, 6 8 , 76, 109-112, 357-365, 376-380 (section I) are particularly useful. * * A t th e en d o f this b o o k th e n u m b e rs o f th e se ex e rc ise s a re g iv e n a p p e a r in th ^ se c o n d e d itio n o f th e “ P ro b lem B o o k ” b y B .P . D e m id o v ie h .

as they;-

CHAPTER V C O N TIN U O U S FUNCTIONS § 20.

Definition of Continuity

A fter the prelim inaries we can now study the m ain problem of m ath em atical analysis, viz* the problem of functional dependence. B ut even now we m ust a p p ro ach our subject system atically and isolate theoretically and practically some classes of functions which are of fu n d am en tal im portance. K eeping in m ind the history of developm ent of our science it is advisable to consider a t the beginning the class o f continuous functions. T h e concept of continuity, i.e. the continuous change of a function, can readily be visualized an d we have already used this term on several occasions w ithout having defined its m eaning. W e m ust now clearly define the concept of continuity an d study the properties of continuous functions in detail, not only because we shall often encounter these functions in future b u t also because the study of other, m ore com plicated classes of functional relationships can frequently be reduced to the study o f continuous functions. L et y = f { x ) be a function w hich is defined along some section o f the n u m b er line an d let a be an a rb itra ry point on th a t line. T h e function f { x ) has a definite value f { a ) a t the point a. L et us now go from the p o in t a to an o th er adjacen t p o in t a + h, w here h is a positive or negative n u m b er w ith very sm all absolute value. In connection w ith this type of transition it is custom ary to say th a t the q u an tity „r whose value is a has received an increment h an d thus taken a new value a + A; we have already said th a t the in crem en t h can be either positive or negative. A new value f { a + h) of the function f (*) corresponds to the new value a + h o f x ; the difference j { a + h) — / (a) w hich corresponds to the difference betw een the new and old values o f y is n atu rally said to be the increment o f y which)

79

•80

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

it has received in the transition of x from the old value a to the new value a + h ; it is obvious th a t this increm ent can be either positive or negative (sometimes it m ay be zero). In analysis it is custom ary to denote the increm ent received by a q u an tity u by the sym bol Aw. YVe can therefore say th a t if we have x — a, th en the increm ent Ay — f { a + h) — / ( a ) o f y corresponds to the increm ent A x = h •of*. Its geom etrical m eaning is represented in Fig. 10. I f a rem ains unchanged while we change the increm ent h of x , then evidently the increm ent A y = f { a -|- h) — f ( a ) of y will also change ; a definite value o f A v corresponds to each value of A x . L et us assume th a t in a p a rtic u la r case the value of h tends to zero, i. e. we assume th a t the new value o f a - \ - h o f x tends to the old value a ; if u n d er these circum stances the increm ent A y o f the function y — f { x ) also tends to zero, it w ould m ean th a t for a sufficiently small change in the value of ,vthe q u an tity y will also change by as little as we please. T his is the m eaning of physical representation o f the concept o f continuity. H ence the essence of the concept o f continuity is the fact th a t an in fin itely s m a l l increment o f the f u n c t i o n corresponds to an in fin itely s m a l l increment o f the in dependent v a r ia b le . Since the relation A y = f(a +

h)

—f(a)

0

(A x

— h

-> 0)

is equivalent to the relation f{a +

h)

f { a)

f^ O ),

the definition of continuity can be form u lated as follows : The function f { x ) is said to be a continuous function at x ~ a (or (cat the point a”) i f f ( a + h) ~ > f ( a )

(h

0

).

H ence it is necessary and sufficient for the function f ( x ) to be continuous a t the point a th a t the value o f the function f (x) should tend to a definite lim it w hen -> a, a n d th a t this lim it should be •equal to the value f ( a ) of this function a t the p o in t a. At the sam e tim e it is also im p o rtan t th a t the relatio n f (a

CONTINUOUS FUNCTIONS

81

should hold irrespective o f the path by which h approaches zero : by positive values, by negative values or w ith o u t a change of sign taking p la ce ; in other words, we should have f ( x ) —>-f(a) irrespective of w h ether at approaches the p o in t a from right or left or w hether it passes repeatedly from rig h t to left an d vice versa (the two-sided lim it o f a function, c f. § 15). T h e precise definition o f the concept o f lim it transitions which we have studied in detail in § 14 enables us to define the concept of co n tinuity in an o th er w ay w hich is often very convenient: the function f (x) is said to be continuous at the point a i f no matter how small s > 0 there is a 5 > 0 such that we have | f (a -+* h) — f {a) | < £ fo r every h whose absolute value is smaller than 5. In o th er words, a function is continuous a t a given point if a change o f th e function w hich can be as sm all as we please corresponds to a sufficiently sm all change of th e arg u m en t. T h e m ajority o f cases in w hich continuity of a function is violated a t some p oint is due to the fact (fig. 1 1 ) th a t f (x) tends to a definite lim it w hen x approaches a from rig h t (h > 0 ) an d tends to an o th er ; definite lim it w hen a: approaches a from left (h < 0 ) b u t these two lim its do not coincide. In this case there is no single lim it lim f (x) a n d the function f (at) is discontinuous a t the point a as can be readily seen from fig. 11. T h e fact th a t a; tends to a from rig h t (i.e. b y assum ing values g re a ter th a n a only) is usually Fig. 11 d enoted sym bolically as fo llo w s: x -> a + 0 ; if in this process / (x) tends to a definite lim it, then this lim it is d enoted by / (a + 0 ) so th a t /(a +

0

) = lim

f(x).

x —» a + 0

T h e m eaning of the symbols x -> a — 0 is sim ilar and f(a —

0

) = lim x

/(* ).

a —0

In the case considered in fig. 11 b o th lim its f (a -f~ 0) an d / (a — 0) exist b u t differ from one another. W e know th a t it is necessary for th e fu n c tio n /(a :) to be continuous a t the point a th a t not only th e

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS

82

limits should coincide b u t also each lim it should coincide w ith the value / (a) of the function’/ (x ) a t the point a (it is evident from th eir definition th a t the n u m b ers/ (a + 0 ) an d / (a — 0 ) m ust not nece­ ssarily coincide w i t h / (a) an d are q u ite in d ep en d en t of it) . T h u s a p a rt from the condition / (a + 0 ) / / (a — 0 ) w hich causes viola­ tion of continuity in the exam ple we are considering, this phenom enon m ay also be due to the influence of o ther causes, viz.: (1) / (a 0) o r / (a — 0) m ay not exist a t all. examples of this kind are the following :

(a)

/ M

— x

(x ^

0

),

0

(,v =

0

);

T ypical

/(.y) increases indefinitely for x —> -f- 0 ; the absolute value o f / (y), which is negative, increases indefinitely for a: -> — 0 ; therefore / ( -f- 0) and/ ( — 0) do not exist; the function / (.v) is thus unbounded in the neighbourhood of the point 0. (b)

sin —

(y === 0),

0

(* =

x

f M

0

);

/ (y) remains bounded this time for y -f 0 ( | sin 1 j x | < 1 ), but cannot tend to a limit, for it repeatedly takes the values + 1 and — 1 (or any value between + 1 and — 1); / ( y ) evidently behaves similarly for y — 0; therefore/ ( -f 0) and / ( — 0) do not also exist in this case although the function / (v) remains bounded in the neighbourhood of the point 0. (2) I t m ay h ap p en th a t/ (a -f 0) an d / (a — 0) exist an d are equal to one an o th er b u t differ from / (a) ; for exam ple /(* ) =

{

1

(y ^ 0), (y = 0);

here / (a + 0) = / {a — 0) for a = 0 while f {a) = 1. In all these cases th e fu n ctio n / (y) is discontinuous (non-continuous) a t y = a (at the point a). I t is very im p o rtan t to rem em ber th a t the definition o f conti­ nuity implies a local (i.e. a t a given point) p ro p erty o f a fu n ctio n ; generally speaking, a function m ay possess this p ro p erty a t some points while it does not possess it a t o th er points. T h e values o f the

“CONTINUOUS FUNCTIONS

83

v a riab le .v for w hich the function f \x) is continuous are known as points o f continuity of this function an d points a t w hich the function is discontinuous as points o f discontinuity. In the exam ples of discontinuous functions, w hich we have considered above, the function is continuous everyw here except a t a single p o in t; the set of points o f discontinuity o f such fu nctions evidently consists of this single point. I t is quite easy to th in k of functions w hich have two, three, or m ore points of discontinuity an d also of functions w hich have an infinite n u m b er of points of discontinuity. But, on the other hand, there are functions * w hich do not have points o f continuity a t all and points of disconti­ n uity fill the entire n u m b e r line. A n exam ple of this kind is given by the function D (,v) w hich we have considered in § 4 ; this function is equal to zero or u n ity in relation to w hether x is an irratio n al or ratio n al num ber. O w ing to the fact th a t every section of the n u m ber line contains an infinite n u m b er of both ra tio n a l a n d irratio n al num bers, no m a tte r w h a t the n u m b er a be, there will be bo th ratio n al an d irratio n al num bers in its im m ediate neighbourhood : hence the function D (x) will take the value 0 and u nity a t points w hich can be as close as we please to the p o in t a ; it thus follows th a t D (.v) c an n o t ten d to a lim it as x -> a an d it is therefore dis­ continuous a t .v = a ; and ow ing to the fact th a t a is a rb itra ry , the fu n ction D fiv) is discontinuous everyw here; m oreover, D (a; -{- 0 )a n d D (.v — 0) do not exist for the sam e value o f x. I t is sometimes useful to distinguish the one-sided continuity of a function. T h e function f ( x ) is continuous to right, o f the point a if / (a -f- 0 ) exists a n d / (a - f 0 ) = f (a) ; it is continuous to left i f / (a — 0 ) exists and f (a — 0 ) = / (a) ; in order th a t the function should be continuous a t the p oint a it is evidently necessary an d sufficient th a t it should be continuous to rig h t as well as left of th a t point. W e shall say th a t the function f (x ) is continuous along the line {a, b) if it is continuous a t every point of this line (i.e. it has no points of discontinuity on this line). In this case continuity to right -of the end a a n d continuity to left o f the end b o f this line are only necessary; this fact is obvious because the function / (.v) is frequently defined only for points on the line (a, b) so th a t the question o f its co n tinuity to left of the point a (or its continuity to rig h t o f the point b) does not arise. T h e definition of a continuous function along a line does n o t a lte r ou r previous statem en t th a t continuity is a local p ro p erty , for continuity along a line is defined in term s o f continuity a t every p o int an d this is the p rim ary definition in the theory of con tinuous functions, w hich implies a well-defined local ch aracter.

84

A C O U R S E O F M A T H E M A T IC A L A N A LY SIS § 21. O perations with continuous functions

In ch ap ter 2 we have studied the results of arith m etical o p e ra ­ tions w ith infinitely small quantities, infinitely large q uantities a n d quantities w hich tend to lim its; we m ust now establish the fact th a t continuity of a function is, as a rule, conserved in the course o f ele­ m entary arithm etical operations. T h e im portance o f this p ro b lem is self-evident, for its general solution will m ake it unnecessary to test continuity of every function obtained as a result o f sim ilar operationsw ith continuous functions. L et us assume th a t we are given the algebraical sum /(* ) ~ f \ M i

/2

M ± ••• =t/*» (•*)

o f functions, each of w hich is continuous a t x — a. A ccording to thedefinition of continuity this m eans th a t ( f n (a) f°r x -► a ; b u t in this case w e know from* theorem 1 § 1 1 th a t

f ( x ) = / i (*) i / 2 (*) rh ••• ±/*» W

(a) ± f s (a) rb •

... ± f n (a) = / ( f l ) for x a and this means that the fu n c tio n / (a:) is continuous at the point a. By using a sim ilar simple arg u m en t (w ith reference to theorem 2 § 1 1 ) it can be readily proved th a t the p ro d u c t of an a rb itra ry constant num ber of functions w hich are continuous at the point a will also be continuous a t th a t p o in t; in p a rticu lar, if the function / (*) is continuous a t * = ■f x (a) a n d f % (x) —>f 2 (a) for a; — a ; therefore it follows from theorem 7 § 1 1 th a t lim A M x->a f 2 (*)

l i m / j (x) x —> a lim f t \ x ) x a

7 1 (a) f 2 (fl) *

w hich shows th a t the fu n c tio n /j (a:) / / 2 (a:) is continuous a t x = a; H ence this rule applies provided t h a t / 2 (a) 9 ^ 0. But if / 2 (a) == 0, then the expression ( a ) / / 2 (a:) is devoid of m eaning for x = a an d therefore continuity of the quotient has no m eaning a t all in this case.

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85

All the rules w hich we have so far established rem ain valid if we consider a function w hich is continuous not at a single point b u t alo n g the entire section o f a lin e ; this follows directly from the defi­ n itio n , given in § 20, of continuity of a function along a section. In the case o f a quotient the result should evidently be stated as follows: if the functions ( x ) and f 2 (#) are continuous along a section of a line an d if f 2 (#) does not vanish a t any point on th a t line, then the function f x (#) ( f 2 (x ) is continuous along th a t line. § 22. Continuity of a composite function Letjy be a function o f x , y = f ( x ) , defined along some section o f a line (a, b). L et us denote by M the set o f num bers w hich the function f ( x ) assumes by run n in g throu g h all the num bers on the line (dy b). L et some third qu an tity £ be an o th er function of y, z — (jy), w hich is defined for all values o fj; belonging to the set M. W hen x takes a definite num erical value on the line [a} b)y then y = f ( x ) also takes a definite value w hich belongs to the set M ; b u t in this case Z = ^ ( y ) also takes a definite num erical value. H ence in the long ru n a definite value of £ corresponds to each value o f a; along th e section (a, b ) ; in other words, z is a function of .v defined along the section (ay b). It is convenient to denote this dependence as follows.’ Z = 9 [/(* ) L ’ o r by two equations : z =--