Advanced Calculus [Second ed.]

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Advanced Calculus

Advanced Calculus Second Edition

WATSON FULKS University of Colorado

John Wiley

& Sons, Inc., New York

•

London Sydney Toronto •

•

Copyright

©

1961, 1969 by

John Wiley

&

Sons, Inc.

No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine All Rights reserved.

language without the written permission of the publisher. 10

9

8

7

6

Library of Congress Catalog Card

SBN 471

Number :7 1-76055

28612 5

Printed in the United States of America

Comments on

the

Second Edition

No

major changes from the

first

edition have been

made

in preparing this

second edition. Misprints and errors have been corrected and passages have

been rewritten to improve the exposition. Exercises, have

Many

problems, especially the

A

been redone.

The most significant substantive changes are now mentioned. The discussion of the number system is somewhat expanded. The harder parts of the discussion of limits and continuity, previously postponed to Chapter 6, have been incorporated in Chapter 3. The Heine-Borel theorem both one- and H-dimensional has been included. Sections on the improvement of coninfinite series, both for series of constants and for Fourier vergence of an

—

—

series,

have been added. Finally, the treatment of Fourier

series

has been

generalized to cover improperly square integrable functions. I offer

my

thanks to

all

attention to errors in the

those

first

who took

edition,

and

the time

and trouble to

especially to

and Hugh Maynard who helped me with the preparation of this new

Boulder, Colorado

April 18, 1969

call

my

Harold Finkelstein edition.

Watson Fulks

to the Student

In using this book, you should give

first

attention to the definitions, for they

describe the terminology of mathematics;

and you can no more hope

to

understand mathematics without learning the voacbulary than you can hope to understand any other foreign language without learning

As with any

vocabulary.

other language, mathematics grows partly by acquiring terms

from neighboring languages

(in this case primarily English),

meanings, and appropriating them into

of the subject. Since mathematics, in language,

its

it is

itself.

modifying their

These form the technical terms

this country, is

imbedded

in the English

important to distinguish between technical and nontechnical

usage.

A

word or phrase being defined

is

set in

boldface type, as for example

uniform convergence. Theorems which have names, such as the fundamental

theorem of calculus, equal sign (=)

is

will

have these names

used in two ways.

x2 It will also

mean

means

=

set in boldface.

mean

A

triple-barred

identity, as for instance

-y 2 = (x- y)(x + y).

" defines " or "

f(x)

It will

2x

2

is

+

3

defined by."

For example,

{-1^jc^7}

that the function /is being defined in the designated interval. There

I believe, little

is,

cause for confusion arising from the ambiguous uses here

indicated. vii

To

viii

the Student

In the body of the

For

text,

occur. Unless a section

found

A

occasional reference

is

made to

exercises

by number.

instance, parenthetical remarks such as (see Exercise B7) frequently

in the first set

number

is

indicated, the exercise in question will be

of exercises following the reference.

few more words about study:

in trying to

understand a proof, you

You

should try to analyze the

should look for the driving force behind

it.

proof and decide that a certain idea (or ideas) involved proof turns on

whose presence It is

this, is

is critical,

that the

and that the remaining parts are secondary calculations

clearly

understood once the significant point

is

seen.

desirable to understand the role each proposition plays in the con-

struction of a chapter.

Which

ones,

of preliminary or secondary

you should ask

results,

which are anticlimactic ? The names

yourself, are in the nature

which are of prime importance, and

—lemma,

theorem, corollary

—provide

a rough rule of thumb, but by no means an accurate one. Finally, after all this talk, let

it

be said that no amount of directions on

how

to study can begin to replace a real and growing interest in the material.

Watson Fulks

to the Teacher

This book

is

designed to serve as an introduction to analysis.

have made an

To

this

end

I

backed by geometrical

effort to present analytical proofs

and to place a minimum of reliance on geometrical arguments. In some stress is laid on this in the body of the text. I have not succeeded completely in avoiding some essential use of intuition in the proofs, but I intuition,

fact,

have localized

my

serious transgressions to Chapter 13 wherein are located

Green's, Gauss', and Stokes' theorems and

Like most

my

action.

who

My

defense

would involve more

some of their consequences.

from the narrow path of

stray

is

virtue, I seek to rationalize

simply that to avoid such geometrical arguments

difficult

work than

I feel is

advisable. In attempting to

hold to a course between the heuristics of elementary calculus on the one hand, and the rigor of function theory and topology on the other,

chosen to err in the direction of heuristics in Chapter this

is,

of course, that what

I sacrifice

in logic I

hope

13.

I

have

The motivation

for

to gain in pedagogy.

So much for apologies.

The book

is

divided into three parts, beginning with calculus of functions

of a single real variable. In aspects of the theory.

this part I delineate

The most novel

feature

some of the more is

integral of a continuous function without the use of

makes ties

significant

perhaps the proof of the

uniform continuity. This

possible postponement of the discussion of the

more

difficult

proper-

of continuous functions until the students have gained some maturity. ix

x

To

the Teacher

The middle

part

is

begins with a chapter

concerned with the calculus of several variables.

on

notation consistently and,

It

and uses vector terminology and vector hope, effectively. Chapter 1 1 which is concerned

vectors, I

,

with inverse functions and transformations, contains a proof of the inversion

theorem which

an adaptation of a new proof by Professor H. Yamabe

is

{American Mathematical Monthly LXIV, 1957, pages 725-726), a proof which

can be carried over to certain types of

The

third

and

last

part of the

book

infinite is

dimensional spaces.

concerned with a treatment of the

theory of convergence as applied to infinite series and improper integrals.

The Cauchy criterion forms the basis for a unified treatment of these topics. The problems in each set are classified as A, B, or C Exercises. This classification is to

problems. The

be taken as an approximate grading of the

A

though the computations are

difficulty

of the

Exercises are straightforward applications of the theory, in

somewhat deeper, and

some of them may be a little

the

C

long.

The B

Exercises

Exercises in general are intended to challenge

the better students.

present

I

influence of I

no bibliography, though I must of course acknowledge the books, old and new, upon the shape this one has assumed.

many

gratefully

acknowledge the influence of colleagues with

the manuscript. In particular,

with

who

whom

I

held a

read critically

I

whom I

discussed

must mention Professor Warren Stenberg,

number of conversations and Professor Jack ;

Indritz,

large portions of the manuscript.

Watson Fulks

Contents

PART

CALCULUS OF ONE VARIABLE

I

Chapter

1

The Number System

1.5

The Peano axioms Rational numbers and arithmetic The real numbers completeness Geometry and the number system Bounded sets

1.6

Some

1.7

Absolute value

1.1

1.2 1.3 1

.4

:

Chapter 2 2.1

points of logic

Functions, Sequences, and Limits

1

3 3 5 8 12

14

16

77

22

Mappings, functions, and sequences

22

26

2.2

Limits

2.3

Operations with limits (sequences)

33

2.4

Limits of functions

39

2.5

Operations with limits (functions)

44

2.6

Monotone sequences Monotone functions

48

2.7

50

Contents

xii

Continuity, Differentiability, and

Chapter 3

More

Limits

Uniform continuity

54 54

3.1

Continuity.

3.2

Operations with continuous functions

58

3.3

The intermediate-value property

59

3.4

Inverse functions

61

3.5

Cluster points. Accumulation points

67

3.6

The Cauchy

71

3.7

Limit superior and limit inferior

76

3.8

Deeper properties of continuous functions

79

3.9

The derivative. Chain rule The mean- value theorem

83

3.10

Chapter 4

criterion

Integration

88

99

99

4.1

Introduction

4.2

Preliminary lemmas

101

4.3

The Riemann

107

4.4

Properties of the definite integral

116

4.5

The fundamental theorem of calculus

727

4.6

Further properties of integrals

124

The Elementary Transcendental Functions

Chapter 5 5.1

5.2 5.3

integral

The logarithm The exponential function The circular functions

Chapter 6

Properties of Differential Functions

129 129

132 136 148

6.1

The Cauchy mean- value theorem

148

6.2

L'Hospital's rule

150

6.3

Taylor's formula with remainder

755

6.4

Extreme values

161

PART

VECTOR CALCULUS

167

Vectors and Curves

169

7.1

Introduction and definitions

169

7.2

Vector multiplications

775

182

II

Chapter 7

7.3

The

1A

Linear independence. Bases. Orientation

186

7.5

Vector analytic geometry

190

triple

products

Contents 7.6

Vector spaces of other dimensions:

1.1

Vector functions. Curves

7.8

Rectifiable curves

7.9

Differentiate curves

Chapter 8 8.1

8.2

En

192

196

and arc length

198 201

Functions of Several Variables. Limits and Continuity

A little A little

xiii

topology: open and closed sets

more topology: sequences, cluster accumulation points, Cauchy criterion

210 210

values,

214

8.3

Limits

220

8.4

Vector functions of a vector

223

8.5

Operations with limits

226

8.6

Continuity

227

8.7

Geometrical picture of a function

230

Differentiable Functions

234

Chapter 9

234

9.1

Partial derivatives

9.2

Differentiability. Total differentials

9.3

The gradient

vector.

The

241

del operator. Directional

248

derivatives 9.4

Composite functions. The chain

9.5

The mean-value theorem and Taylor's theorem

9.6

The divergence and

251

rule

for

259

several variables

Chapter 10 10.1

curl of a vector field

Transformations and Implicit Functions. Extreme Values

Transformations. Inverse transformations

261

267

267

10.2

Linear transformations

269

10.3

The

276

inversion theorem

10.4

Global inverses

284

10.5

Curvilinear coordinates

285

10.6

Implicit functions

289

10.7

Extreme values

294

10.8

Extreme values under constraints

297

Chapter 11

303

Multiple Integrals

11.1

Integrals over rectangles

303

11.2

Properties of the integral. Classes of integrable functions

310

11.3

Iterated integrals

311

11.4

Integration over regions. Area and

volume

316

xiv

Contents

Line and Surface Integrals

Chapter 12

325

12.1

Line integrals. Potentials

325

12.2

Green's theorem

336

12.3

Surfaces.

12.4

Surface integrals.

12.5

Stokes' theorem. Orientable surfaces

357

12.6

Some

363

12.7

Change of variables

PART

Area

345

The divergence theorem

351

physical heuristics

365

in multiple integrals

THEORY OF CONVERGENCE

111

Chapter 13

373

375

Infinite Series

13.1

Convergence, absolute and conditional

13.2

Series with non-negative terms:

13.3

Series with non-negative terms. Ratio

Comparison

375 tests

and root

tests.

379

386

Remainders 13.4

Series with variable signs

389

13.5

More

392

delicate tests

13.6

Rearrangements

394

13.7

Improvement of convergence

400

Chapter 14

Sequence and Series of Functions. Uniform Convergence

408

408

14.1

Introduction

14.2

Uniform convergence

409

14.3

Consequences of uniform convergence

415

14 .4

Abel's and Dirichlet's tests

423

14.5

A

427

theorem of Dini

Chapter 15 15.1

The Taylor

Power

Series

series. Interval

of convergence

430 430

436

15.4

Properties of power series The Taylor and Maclaurin series The arithmetic of power series

15.5

Substitution and inversion

455

15.6

Complex

457

15.7

Real analytic functions

15.2 15.3

series

442

447

460

Contents

Improper Integrals

Chapter 16 16.1

Improper

xv

463

Conditional and absolute

integrals.

convergence

463

16.2

Improper

471

integrals with non-negative integrands

16.3

The Cauchy

16.4

An

16.5

Improper multiple

474

principal value

475

alternation test

Chapter 17

integrals

477

-

Integral Representations of Functions

483

17.1

Introduction. Proper integrals

483

17.2

Uniform convergence

487

17.3

Consequences of uniform convergence

493

Gamma

Chapter 18

and Beta Functions. Laplace's Method and Formula

Stirling's

509 509

18.3

The gamma function The beta function Laplace's method

18.4

Stirling's

18.1

18.2

Chapter 19

512 515

formula

520 523

Fourier Series

19.1

Introduction

19.2

The

class

ffl 2

523 •

Approximation

in the

mean.

Bessel's

529

inequality 19.3

Some

19.4

Convergence theorems

19.5

Differentiation

19.6

Sine and cosine series.

19.7

Improvement of convergence

552

19.8

The Fourier

554

19.9

Function spaces. Complete orthonormal

useful

lemmas and

532

536

integration.

Uniform convergence

Change of

544 548

scale

integral sets

560

Elementary Differentiation and Integration Formulas

567

Answers, Hints, and Solutions

569

Index

593

Advanced Calculus

PARTI

CALCULUS OF ONE VARIABLE

CHAPTER

1

The Number System

The Peano Axioms

/./

Any is

study of mathematical analysis has

its

basis in the

number

therefore important for students of analysis to understand

metic can be developed from the natural numbers (another positive integers). It

here however, ;

is

how

system.

It

the arith-

name

for the

not our intention to carry out such a development

we do want to make some comments about the logical structure

of that development.

A

standard starting point in the development of the real numbers

certain set of axioms that were

first

is

a

formulated by the Italian mathematician

Peano. These axioms state that the natural numbers

satisfy certain properties.

From only these properties, making appropriate definitions as we proceed, we can develop all the usual rules of arithmetic. In the terminology of formal logic, we have a set of undefined objects that we choose to name the natural numbers, satisfying Peano's axioms. This means merely that the natural

numbers are taken terms of which

as the basic

"atoms" of our mathematical system

in

we express the other mathematical concepts, but which are

themselves not expressed in more fundamental terms.

Perhaps an easier way to visualize the situation, since after

all

claim that you never heard of arithmetic or rational numbers,

you cannot is this

can verify that the positive integers are a system of objects that do

:

We

satisfy the

5

The Number System

4

Peano axioms.

We

can

now

proceed to recapture

all

we know of arithmetic,

using only those properties stated explicitly in the axioms.

The

five

Peano axioms

follow.

Axiom

1.

Axiom

2.

Axiom

3.

The number

Axiom

4.

If two natural numbers have equal successors, they are themselves

a natural number.

is

I

To every natural number n there is associated another natural number n' called the successor of n.

equal. That is,ifn'

Axiom

= m\

1 is not

Suppose that

5.

=

then n

M

is

in

a unique way

a successor of any natural number.

m.

a collection or set of natural numbers with the

properties 1 is in

(i)

Then

M,

n' is in

(ii)

M whenever n M

the collection

is in

consists

M.

of all natural numbers.

Here equality is used in the sense of numerical identity that is, m = n means that m and n are symbols standing for the same number. Thus we ;

take (i) (ii) (iii)

m — m, m = n implies that n = m, m = n, n = k implies that m = k,

as part of the underlying logic

and do not

list

these as part of our set of

numerical axioms.

Notice that

this list

of properties does not contain any explicit statement

of the linear ordering of numbers. There tinguished by not being a successor. turns out to be n the

way

addition

numbers must 2, 3, 4, 2'

.

.

+ is

line

1

after addition has

defined.

up

after

We 1

is

a "first" one, namely

The successor

n'

been defined. In fact

this is essentially

can then proceed to show that the natural

in their familiar order:

are not given in the axioms, they are defined

.

dis-

1,

of a natural number n

(Since

1, 2, 3, 4,

by 2

=V=1+

1,

3

=

= 2+1,....) Axiom

5 is precisely the principle

of mathematical induction.

It

is

an

important tool in the technical development of the properties of the number

we have been discussing. We shall have occasional use for this principle later, and we shall see that the statement of Axiom 5 is equivalent to the more familiar one given in elementary texts. As indicated earlier, we can start with the natural numbers and, knowing system that

only the properties and relationships stated by the axioms, derive arithmetic.

Why

do you suppose mathematicians

all

of

are interested in starting

Rational Numbers and Arithmetic

m+

from such primitive beginnings ? Are

n

=

n

+ m and m-

truer because they can be derived as a consequence of the sense, quite the contrary

is

from them we can derive

The axioms

true.

m+n=n+m

addition and multiplication), and

One

=

n-

m

any

axioms ? In one

are "true" or valid because

m

and

•

n

=

m

n

(after defining

the other standard rules of arithmetic.

all

aspect of such an approach

n

5

is

aesthetic:

it

gives a clean, beautiful

development of a well-known and useful body of knowledge from simple, precisely stated foundations.

Another somewhat more pragmatic

What

we

result is that

it

serves to help answer

do mathematics without a precise foundation, a "proof" may be any convincing argument, based on what one " knows." But since this knowledge is not well formulated, this may lead to the question,

a proof? If

is

try to

Theorem A may be based eventually on Theorem B, Theorem B may be based on Theorem C, and Theorem C eventually based on circular arguments:

A

again. If the proofs are long

and involved,

it

may

be

spot such

difficult to

Having a well-formulated axiom system reduces the of such mistakes. A proof becomes a logical argument based

circular arguments. possibility

on the axioms, or on previous theorems based on the axioms. Of

directly

course there

is

always the possibility of

human

error, for

everyone makes

mistakes but at least this approach gives a clearer idea of what ;

we

seek in a

proof.

In any case, this

is

an example of what

the study of mathematics.

By

this

is

called the axiomatic approach to

we mean merely

that the whole of a certain

section of mathematics has been reduced to a few statements, plete structure

and the com-

can then be unfolded from these by a study of the logical

consequences of these statements.

Rational Numbers and Arithmetic

1.2

After the salient properties of the natural numbers have been developed, the next step

numbers, you

is

the introduction and study of rational numbers. Rational

will recall, are those

numbers

that can be represented as ratios

of whole numbers; in other words, rational numbers are

We list

common

below certain properties of rationals that can be deduced

fractions.

in the

course

of their development from the natural numbers. Thus these properties of rational

The So a

>

numbers are

actually consequences of the

Peano axioms.

read "

is

greater than," precludes the possibility of equality.

b means that a

is

greater than b. Thus, in particular, a

sign

>

,

is

not equal to

:

:

The Number System

6 b.

The




where, of course, the pi's need not be distinct. This, however, tion to the hypothesis, since n 1.3 b

There

Corollary.

You

are,

as points

is

is

is

a contradic-

expressed as the square of a natural number. |*

no rational number whose square

is 2.

of course, familiar with the geometrical interpretation of numbers

on a

line.

For the purpose of emphasizing certain aspects of

this

we review its main features here. On a straight line we choose and call the origin. We take a convenient unit of a point that we mark length, and to each number x we make correspond the point whose distance from is x measured to the right if x is positive and to the left if x is negative.

interpretation,

Let us note that the arithmetical law

a> finds 1

,

Pz on a

then Pi

Now,

b

>

P

line: If

to the right of

lies

given any rational

P3

to x. Let

U contain

and

number

of

x, the set

all

P2

to the right of

second class

number x

number x

L and

each

R contain all

U

first class

member of L

P on

x and

let

L

contain

itself in either

is

all

the

U or L but not

is less

R

into

classes in

than every

and

L

all

(right

member of

U.

points of the line can be

and

left)

with respect to

points to the right of P and

of P, with the point

let L contain all points to the P itself in either R or L but not in both. For all points

the separation or cut leads to a partition of the line into

that each point in the

L and

two

greater than every element of the

the line, the set of

separated into two subsets or classes

Now

,

and lower) with respect

numbers

divides the rational

the

Similarly for any point

class

P3

numbers can be

rational

U and L (upper

the rationals greater than

all

which each member of

P

c

Such a separation or cut of the rational numbers associated with a

given rational

left

>

.

rationals less than x, with the rational

P. Let

P2

to the right of

lies

t

separated into two subsets or classes

in both.

a

implies

c

geometrical counterpart in the following statement about points

its

P ,P2

b,

first class

each point in

there are points

we mark

L is

R

is

then to the

on the

two

classes such

to the right of every point in the second

fine that

left

of every point in R.

do not correspond

whose distance from

to any rational

we

see

by

Corollary 1.3b that there can be no rational number corresponding to

it.

number. For In

fact, it is

if

not very

that correspond to *

The symbol |

no

will

the point

difficult to

rational

show

is

y/2,

that there are infinitely

number. This means that

be used to denote the end of a proof.

It

if

many

we mark

all

replaces Q.E.D.

points

points

The Real Numbers: Completeness

11

on the line that correspond to rational numbers, there are infinitely many unmarked points left over. Actually, in every interval of the line there are points corresponding to rational numbers and points that correspond to no numbers

rational

(see Exercises

C3 and C4

in Section 1.7).

These considerations lead to the recognition of the existence of gaps or a lack of continuity or completeness in the distribution of rational numbers as

compared with the

on a

distribution of points

line.

We

accept as a basic fact

of geometry that points are continuously distributed on a there are

but take

no gaps or incompletenesses it

in the line.

For

this

line; that

we

offer

is,

that

no proof,

as a geometric axiom.

Although we

no proof of this concept of the continuous distribution of points on a line, we do want to formulate it in a precise form that we can use in our discussion of the number system. The formulation we use is due to the

offer

German mathematician Dedekind, whose

closely.

treatment

Dedekind uses the separation or cut described

the completeness of the line. His formulation

Suppose that

all

is

we

are following

earlier to characterize

as follows.

points on the line are separated into two

nonempty

classes

R is to the right of every point in L. Then there is exactly one point P that produces this cut, and P itself is either the rightmost

R

and

L

so that each point in

point of L or the leftmost point of R.

We offer no proof for this statement, any more than we do for the statement on the line. We take this to be merely

that points are continuously distributed

a precise statement of that continuity or completeness, a formulation that

adapted for the purpose of considering the analogous questions for the

is

dis-

tribution of numbers.

The analogous statement for rational numbers cannot be true. To see this, we need only give one example where it fails, and this example we now describe.

Let the upper class U, corresponding to positive rationals

other rationals. Suppose that the cut

Then

it is

R

for the line, be the set of

whose squares are greater than

possible to

show

is

1

.3b.

of the completeness axiom of the

and

let

L

be

all

all

the

produced by a rational number

(the technical

But this contradicts Corollary

2,

proof will be

omitted) that x = 2

x. 2.

Thus we see that the analogue for rationals,

line, is

not correct.

The final big step in the development of the number system is to define numbers to fill the gaps displayed by the previous remarks. These new numbers are called irrational numbers. The system of numbers consisting of the rationals and irrationals together is called the real-number system. Thus a real number can be either rational or irrational. From now on the term "number" will stand for a real number unless otherwise stated.

The Number System

12

The will

details involved in carrying out the definition of the

We

be omitted.

number

want to emphasize two important

facts

new numbers

about the

real-

system. First, the notions of addition and multiplication can be

defined for real numbers so as to satisfy the properties listed for rationals

under

I

and

II

of Section 1.2; that

we read

is, if

"real

number"

in place

of

"rational number," the conclusions stated there remain true. Second, and

very important, the real-number system

is

complete; that

axiom on the completeness of the

the geometrical

line

is,

the analogue of

can be proved as a

theorem.

Dedekind's Theorem.

///.

two nonempty classes

R

number

in

L. Then there

number

in

L and less

We ments

Suppose that

and L, so

all real

number

that each

a real number x that

is

than or equal to every

numbers are divided

in

R

is

greater than or equal to every

is

number

in

R.

designate this by the numeral III for the following reason. listed

section,

under

I

and

II

of Section

1

.2,

The

state-

together with III of the present

can be taken as an axiom system for the description of the

numbers. That

is, all

properties of the real

into

greater than every

real

numbers can be developed from

these statements alone.

The

scientific

outlined here

importance of the development of the real-number system

is,

of course, that

a cleanly laid out, logical form. the analogy between real

1.4

Geometry and

Analysis

is

puts a large body of useful knowledge into

numbers and points on a

removes analysis from any theorems of calculus can

it

We also want to emphasize that it strengthens

logical

now be

the

the

same time

dependence on geometry. All the basic

given purely arithmetical proofs.

Number System

that branch of mathematics that

is

with limiting operations. Mathematicians, as you last section, like to

line, yet at

concerned with

may have

limits

and

surmised in the

base their study of analysis on arithmetic, with no logical

dependence on geometry. However, as you already know, geometrical pictures can be very enlightening when you are attempting to solve many types of

no objection to using such devices; indeed you should be use them. But use them as guides that give insight, and not, if

problems. There

encouraged to

is

you can possibly avoid

it,

as essential parts of an argument.

Geometrical language, with useful

its

highly suggestive power, can also be very

and can be given arithmetical meaning.

We

define a few geometrical

Geometry and

the

Number System

13

terms here, and others occur intermittently throughout the rest of the book. This, of course, out.

is

an attempt to exploit the geometric analogy we have pointed

We thus rely heavily

we avoid any

well, except in

reasonably essential

We

on geometry

essential use of

Chapter

in

our arguments. In

When we do

13.

way, we are careful to point

where possible

for our inspiration, but

geometry

this

we succeed

lean on geometry in any

this out.

proceed to define a number of geometrical terms. Since there are quite

a few of these, you will want to refer back to this section until you become familiar with

We

these terms.

all

speak of the real-number system as a one-dimensional space, and of

course

we

visualize

it

as a line.

In one-dimensional space

A

point set

An

interval

we

use the

word

point to

mean

" number."

a collection of points.

is

is

a point set described by inequalities of one of the following

types

{a {a fa {a

^ ^ <
if

n

we choose

N=

\/e

TV,

2

N

n

This completes the solution.

Example 3 f(x)

We that

= yJx(Jx~+\

assume that for every x

is,

there

proved in a It will

is

>

a unique positive

D

-Jx\

there

:

{x

>

0}

a unique positive square root;

is

number y

which y 2

for

=

x.

This

will

be

later chapter (see Section 3.4).

know

also be convenient to

a

which we leave as an

2

that

> a>

if

>

a

Ja >

1,

1,

exercise.

Now

y*

i '

y/x+l + from which

it

Jx~

y/\+

seems clear that the

(1/JC)

limit

+

is ^.

1

Hence we consider

/(*)-

Vl +

(1/x)

+

yu a/*) +

2

2

1

l+0/x)-l_ Vl+(l/*)-l < 2(1 + 1) 2(7l + (1/x) + 1) So

if

we

take

XXo

be l/4a or larger, and x

/(*>--


i

X, then i

Tx < 4x^

theorem that establishes a

1

4x

e-

limit very useful for

comparison

Exercises 2. 2d

Theorem.

Proof. Let

=

jc

31

Mw^^Kjjjhery^Jm \/a

—

1.

Then x >

and 1

so for n

^

11

1

=

a

(1

where the

first

inequality

\a

n \

is

=

+

x)

n

^

1

+

— nx 1

< nx

by Bernoulli's inequality. Thus

a

n




X8

A 1.

Exercises Determine whether the following sequences have limit

A

\a n

where

a„

is

-A\ N,

(e)

+\

-Y-. rr + n

A

exists

i

COS cos x.

x {d)

2.3

+\

>

^ then a 2

3.

Prove that

4.

Define lim x _»_ „ f(x).

if

a

1,

limits at infinity. In

JC

(c)

(b)

+ 2,-1

n

•••

rr

Jt

1

cos -

0)

> a > Va >

1.

2* 3

+

+

2

3

Jt

+ -• rr

each case

x>X.

if

a:

2

rr

determine an X(e) such that

\f(x)-A\

oo,

and /(x) ^ #0) ^

h(x).

co.

^0,

n 3 /2 n

^0,

n A \2 n ->0.

(//////:

See Example

2.)

Show

that

Operations with Limits {Sequences)

Show

3.

2

n

that n a ->0,

«V^0,

4

and n a"-+0

above and see the proof of Theorem

there

But n k

^

is

We know

be the limit of the sequence {an }.

that for each

N for which

an

A\

>

if

n

if

k

if

k>N.

N.

k; hence

N

>

N.

Thus

-A\ =

\b k

The following

Theorem.

2.3b

a constant

is

Proof. Let a

=

|

are a collection of standard operational theorems with

which you should become

there

-A\N.

(Why?)

Then

\>\an -a\>\an \-\a\.

(Why?)

Thus \an

(1)




-b\.

\a\\b n

e2

and zero.

make

7V2 such that

\a„

-

a\


N

|6„

-

6|


N2

t

and

Thus

ifn>N= max (JVj, 7V2

from

(3)

\a n

From

bn

-

ab\

this proof,

)

(that

is,

^ Me +

|a|e 2

-s

l

is

2(M +

2(| fll

+

1

N

i-

(Why?)

Then

IM> (4)1*1 Hence

for

n> N

if

n>N,.

(Why?)

t

1

1

b„-b

K

~b

Kb


0.

Show by examples Show

+«

i /

(v

is

*2

1

r

(c)

9.

Vln+l-Vn.

.(./rri-,y

Show

8.

(d)

«

/i

+ 3)

3.

7.

1

3

Exercises

that

6.

and Limits

Functions, Sequences,

which a„ k

$s 0.

Do

and the

Theorem

What

{b n } are divergent sequences, then {a„

same

for {an

—b

2.3g remains valid

modification can

now

and

n)

if

be

+ b„}

{a„/b„}.

there

made

is

in

a subsequence

Theorem 2.3h?

39

Limits of Functions

C 1.

Exercises Show

that

if

and

{a n } converges to zero

{b n }

is

bounded, then {a n b„} converges

to zero. 2.

Evaluate the limit of each of the following sequences.

H ((^)"| (b)

*>y>o. aj>0J=l,2,...k.

ful^J],

x^y>0. ( 1.

= 1 + h and use the binomial theorem.) = (1/n) 2" and show that lim u = a.

(Hint: Set ^/*

n

= a.

Show by examples

that the converse of (a)

Set o n

sequences {a„} such that lim a n

2.4

o-i

= a,

n

,

is

not true; that

but lim a„ does not

is,

there are

exist.

Limits of Functions

For some positive h,

let/(x) be defined in the interval /: {a

except possibly at the point a

borhood of

a.

itself.

Then a number

a (or as * approaches a\ [6 of course depends

on

|/(jc)

We

,

Suppose that lim a n

e,

if

That is,/(x)

is

-A\

there

is

a number

frequently written


00

x

[-»©]

Exercises Evaluate

V(a +

bx)(c

+ 0

2.

Evaluate Vim X->

00

Ux-Vx-yJx+ Vx).

— x ^ for all x > 0, If oxy + by ^ for all x,

If

ax

2

a:

2

>>,

show that a < show that a =

Recall (Section 2.1) the definition of

[x].

6

^

.x-o +

0.

that, for

X 6

-, lim

a

0,

Show

6

lim x-0 + a

1.

# x

a>0, b> 0,

= 0.

Discuss the left-hand limits of these functions.

= e llx for x i^ Set /(x) = e llx /(e llx — 1) x/(x) at x = 0. Set/(x)

C

and discuss the one-sided

x

^

Discuss the one-sided limits at

x

for

and

=

limits of /at

of arc tan

1/x.

Exercises

1.

Prove that

2.

Evaluate

[

^j/x ->

1

as

x

-> oo. (See Exercise

C3 of

Section 2.3.)

V(\+aixKl+a 2 x)---(\+anX)-\ lim x-0

x

— 0.

discuss the one-sided limits of

/and

of

48 3.

Functions, Sequences,

Let x a be the root of ax 2

and Limits

+ bx + c = that has the smaller absolute value. = — c/6 = the root of the limit equation.

6t^0, then \\rc\ a ^ x a happens to the other root? that

4.

if

Prove the " converse " of Theorem 2.4a that x„ -+a, then lim x ^

fl

/(» =

:

If /(*„) ->

A

Show What

for every sequence {*„} such

A.

Monotone Sequences

2.6

A very important class of sequences are monotone or monotonic sequences, More

those that steadily increase or decrease.

monotone-increasing

if its

a n + \^an In the that

case

first

we

sequence {an }

is

elements an satisfy an inequality of the form or

an + 1

>a

n

w=l,2,

,

say that the sequence

nondecreasing. In the second case

it is

precisely, a

....

weakly monotone-increasing or

is

we

say that the sequence

is

strictly

increasing. Monotone-decreasing or nonincreasing sequences are defined

by

reversing the inequalities.

In particular, it

if

a sequence

is strictly

increasing

weakly increasing; but the converse

is

Thus any theorems established

—that

is,

an+1

>

an

—then

of course not necessarily true.

is

for weakly increasing sequences apply to

strictly increasing ones.

We

which

establish the following theorem,

is

a consequence of Theorem

1.5b.

Theorem.

2.6a

Then lim a n

exists

Let {an } be a nondecreasing sequence that

lim an Proof. Since {a n }

by Theorem

1.5b.

want

Let

e

to

>

Hence, for

show

is

=

sup an

a set of numbers that

.

is

that

A

sup an

bounded above, sup an

.

of a

satisfies the definition

be given. By Theorem 1.5a, there

n>

bounded above.

Let

A =

We

is

and

aN

>A —

an

^

aN

A ^

a„

>A -

is

e.

TV,

A ^

>A —

or £,

e,

limit.

an TV for which

exists

Monotone Sequences

49

or, finally,

^A -


N

if

Hence, certainly,

-a \

n

if

n

N.

I

Let {an } be a nonincr easing sequence bounded below,

Then lim an exists and lim an

The proof of

=

an

inf

.

this corollary is similar to that

These simple but important use throughout this book.

of Theorem 2.6a.

powerful tool that we

results supply us with a

As an example,

let

us consider the sequence given

by a„

(You

will see that

series

X?

=l+ we

l

11 - + - =Z "

1

+ 2T+3 + i

o

!

1

^.

are really discussing the convergence of the infinite

1/^9

Clearly

a" +

Hence an+l

show

that

*«

>

an

it is

=

1

,

>a

=

a"

(7TT)!

so that the sequence

monotone-increasing. Let us

is

now

bounded above:

111 + + +-+ -"+ l+-ill + ?+ + +— 1

+ 1+

2!

-4(-3(-D-(«-H

—

a

This, of course,

form from your

is

an idea with which you are already familiar

earliest

in a primitive

concept of a function, because most of the functions

with which you are familiar are continuous, except possibly at a few points.

A function may fail to /may

be continuous at a point a for any of three reasons:

not be defined there; the limit

the limit exists, then the

a.

That

is, it

it may still happen that it is left-continuous may be that

lim /(*)

54

not exist; or if/ is defined at a and

two may not be equal.

If/is not continuous at a point a,

or right-continuous at

may

=

/(«)

55

Continuity. Uniform Continuity

or

lim /(*)

=

/(0.

is,

Clearly /(x)

is

x=

left-continuous at

x=

right-continuous at

we

0. If

limits there,

=

+

1

sgn x (see Section 2.1)

It

.

— 1 and + 1

namely

,

we had taken/(0) = +1, it would be

0. If

take f(0) as anything

x=

neither left nor right continuity at

f(x)

= 0.

0,

although

then f(x) has

else,

has

it

left

and right-hand

respectively. In particular, for the function

we have /(0 -

0)

= - 1,

=

/(0)

0,

/(0

+

0)

should be clear that a necessary and sufficient condition that

continuous at x

=

a

is

that

it

be both

left-

=

/ be^

and right-continuous. (See Theorem

2.4b.)

The

definition of continuity

been defined in for

e,

S terms.

many purposes

A each

depends on the concept of a

We

better to

it is

—

function /, defined in {x e

>

there

a

is

<

jR,

and

let

are so related that

and $[/(*)] = x for every x in D, then $ is The basic inverse-function theorem is the

said to be the inverse function of/.

following.

Theorem.

3.4a


(y).

= y.

3.3b, for

on {a

increasing

then there

< y ^ /?},

each y in {a

continuous function

is

which

U{x)-]

=

(y)

= x.

on

a unique inverse function

/?} j>,

is strictly

there

and

is

will

increasing

exactly one

be denoted

*

62

and More Limits

Continuity, Differentiability,

Taking /of both

sides,

we

get

/WOO] -/(*)Hence the unique function

We yi

can see that

> y>i

>

an d

cj)(y) is

Choose y y and y 2 say Then

increasing, as follows.

is strictly

x t and x 2

let

the inverse of/,

,

respectively, be the corresponding x's.

,

suppose, for contradiction, that

=

*i

Taking /of both

sides,

is

—

^(jo the

•

=J(x )^f(x 2 )=y 2

.

l

we

the continuity,

strict

monotonicity of

.

use a contradiction argument again. Suppose

discontinuous at some point y in {ct^y^p}. Then either 0) < (j)(yo) or (y 2 )

a contradiction, and hence establishes the

To prove that




-

(yo)

For every


x2

;

where b

2

>y

.

(How

byf(x)

= x2

on

can we be sure that

Now on this interval/is strictly increasing:

then

x\

—

x\

= Oq —

x 2 )(x 1

+

x2)

>

0,

x x — x 2 > 0. It is also continuous hence a unique inverse exists. This means that for every y in {0 < y ^ b 2 } there is a unique x such that x 2 = y. since

;

This holds in particular for the given y

By

the

|

.

same type of argument, we can prove the existence of a unique nth

root of any nonnegative number. Since integral powers are also defined,

now

can

define

what we mean by \l~cT

a"

/m

namely ^fa

,

= {^Ja) m

(Exercise

n .

It is

easy to see that

B\ 1).

somewhat more difficult to define a x for arbitrary x. This will be poned to the chapter on the elementary transcendental functions.

It is

A 1.

we

post-

Exercises Determine whether the following functions are continuous indicated. In each case is

where the function

uniformly continuous. (x 2 ,

(a)

fix) (*,

0 n

+ 1 and — 1, e of — 1

l,2,3,....

no one number A such that xn 's do cluster around two distinct

limit, for there is

>

so that for any s

Such a point

.

The

=

w

for large n. But the

and others within

value of the sequence.

+

there are x„'s within e of

precise definition follows.

called a cluster point or cluster value of the sequence {xn } if

is

a subsequence {x„ k } converging to

Similarly, if/ is defined in a deleted

A

:

neighborhood of a, we say that

cluster point or cluster value of /at a if there is a sequence {xn } with

which f(x n ) -> A. For example, the function f(x)

hood of

the origin

sequence {x n }

A

=

and has

{[2/(4«

+

+1

=

sin l/x

is

A

-+

if

A

is

a cluster value

every sequence {x„} with xn -*

is

a

a for

defined in a deleted neighbor-

as a cluster value at 0. If

l)rc]}, this is

xn

we

consider the

immediate.

limit of a sequence or a function is a cluster point or cluster value.

the other hand,

+

called a cluster point or cluster

is

of/

at a, then

a, it is true that

A=

f(xn ) -+A

lim x ^ a f(x)

if

On for

(see Exercise C4,

Section 2.5).

We define a similar concept for sets: set

S

if

point a

a point a

is

an accumulation point of a

every deleted neighborhood of a contains points of S. Note that the itself

may

or

may

not be a

We come to the basic theorem

member of the

on

cluster

set S.

and accumulation

points,

namely

—^—

the Bolzano-Weierstrass theorem.'*

fi

3.5a

Theorem.

A bounded infinite

and a bounded sequence has

set has at least

at least one cluster point.

one accumulation point,

68

and More Limits

Continuity, Differentiability,

We

write out the

sets. The proof for sequences is from the one for sets by replacing such

part of the proof for

first

and can be read

quite similar

off

many members

phrases as "infinitely

S"

of

"

by

an for

many

infinitely

distinct values of w."

S

Proof. Let

be the

S

Since

set.

bounded, there

is

an

is

M such that S

is

contained in the interval

:{-M^x^M}.

I

When we bisect I to form the two closed intervals { — Af^x^O}, {0 ^ x ^ M}, one (or maybe both) of these contains infinitely many members of S", since S is an infinite set. Denote one such interval by I± Bisect It one .

of the two intervals must contain

and continue

it,

intervals I

,

Ix

,

.

,

.

.

,

we denote the end ^ x

ni

for which

\A

—

x n2

\




n2

nk

>

nk _

Pur

next theorem

3.5b

Theorem.

of In tends is

an

is

J

for

which

\A

—

for

which

\A

-

clear that the subsequence {x„

intervals, so that

is

known

Let

In

to zero as

{/„}

at least

in

less

is

infinity, then there is

A and

The requirement /„: (0,

I

can be taken all

in the previous proof,

A and

distinct points,

a fixed positive number

A' are in

If the length

n.

exactly one point that

less

we

see

the intervals.

all

d.

A'.

The

distance

However, the length of /„

than any preassigned positive number,

both

.

t

one such point in

particular, the length

N. There are only number of x's not covered by this bound. Hence, by increasing 1 if such that necessary, we get a number Keep

a

M

finite

M

\xn

Thus

{x„}

is


oo,

we

nk

limit.

a subsequence e/2,

we have

>k>N^j.

get

£

if

F!>Jv(|V

2 This

is

the definition of a limit.

The Cauchy

criterion extends easily to cover the existence of the limit of a

function.

3.6b the case

Theorem.

Let f(x) be defined

of one-sided limits

it

in

a deleted neighborhood of a

need be defined only on one side of a).

and

sufficient condition that \\m x _+ a f(x) exist is that for

3(e)

for which

whenever

\f(x)-f(x')\ 0, there

is

mm

1

an

N

•

•

•

~Xn

)\

-x m _ 2 +

•••

\

+ +

cr r

+

-X

n

\

n

m~n~

l

)

-r such that




there

is

*„

N for

whose points

which x„ lie

above

hence by Theorem 3.5a

But

is finite.

Then for each

£

^A+ A+£ it

>

£

£ .

.

n

if

A

N2

>

.

0,

this

there

is

were not so suppose :

/?,

for

is

also

which

supremum of the

all

is

of

bounded above,

clearly set

ft

^A+£

.

of the cluster

points of {xn }.

I

I

A

A

N there

a subsequence {x„ k },

subsequence

the

is

this

so that for arbitrarily large

has a cluster point

this contradicts the fact that

.

Suppose that

Then

But

t

such that

2 (e)

We prove the first statement.

that there were an exceptional

n>N

if

xn > X — Proof

A

Let {xn } be a sequence for which

an N^e) such that

—e

AA+£ I

I

A

I

glance at the previous figure should convince you of the truth of the

next theorem. 3.7b

Theorem.

\\mxn

In order for

to exist,

it is

necessary and sufficient

that

lim sup

Proof: Necessity. If lim xn consists of exactly

=A

xn = lim exists,

one point, namely

A=

A

X

=

inf xn

.

then the set of cluster points of {xn } itself.

A.

Hence

78

Continuity, Differentiability,

there

is

an

A=

— s< xn < A + -




N2 of that theorem) such that n>

if

supremum of the

n

if

N.

N.

|

lim sup x _* a f(x) or

a,

cluster points

we

a,

>

a point can be defined similarly. Thus,

neighborhood of

in a

any neighborhood of

in

e

e

limit superior of a function at

defined as the

above

and

3.7a, for

can be rewritten as

X, this

/is bounded above is

Then, by Theorem x

\xn

The

X.

N (take N to be the larger of N X

Since

A=

Suppose that

Sufficiency.

and More Limits

of/ at

a. If

/ is

if

Iim^ f(x)

not bounded

say that

lim sup f(x)

= + oo.

x-*a

Similarly

we

define lim infx _ a f(x) or lim x _ /(x) fl

and lim sup x ^ ±o0 f(x) and

liminfx _> ±00 /(x).

The next two theorems proofs are

Theorem.

3.7c

Then for each

are the analogues of the

two previous ones. Their

as exercises (B3).

left

e

>

Let

f be

there

f(x)

^ lim

is

bounded above

in

a deleted neighborhood of

a.

a S^s) such that

sup f(x)

+

£

if


a

and iff is bounded below f(x)

there

is

a S 2 (e) such that

^ lim inf f(x) -

s

if

|/(«-/(«|

=

0,

which contradicts the construction that

l/(xJ-/(OI>«o-

/4

1.

I

Exercises Show

that the following functions are uniformly continuous in the intervals

indicated.

Vx + 2

(a)

.

1,

{O^x^l}.

The Derivative. Chain Rule 1

x2

x

sin

(b)

^ 0,

x

{O^x^l}.

= 0,

jc

1

83

— COS X

x^O,

(c)

x

= 0,

H**4

x^O, id)

x=

[0 (e)

2.

{-l^jt^l}.

* 5 sgnx,

Show

{0^*^!}. 0,

that if f(x)

is

uniformly continuous on an interval

continuous on each subinterval of 3.

Prove Theorem

4.

Let

/ be

1.

it is

uniformly

3.8c. ;

bounded

F is

interval.

Exercise Show

that

-J and f(b

C 1.

then

continuous on a closed bounded interval prove that the range of

also a closed

B

/,

/.

if /is

+ 0)

continuous on an open interval /: {a

both

exist,

< x < b},

then / is uniformly continuous on

and

if f{a

+ 0)

/.

Exercise Give an alternative proof of Theorem 3.8d using the Heine-Borel theorem.

3.9 It is

The Derivative. Chain Rule assumed that you are familiar with elementary

differentiation formulas.

A list of these can be found in the back of the book. Very likely your exposure to formulas involving the logarithm, the exponential,

functions was based on elementary and

of these functions. In Chapter 5

we

somewhat

and the trigonometric

unsatisfactory definitions

define these functions

and put

their

84

Continuity, Differentiability,

and More Limits

on a firmer foundation. In the meantime we

properties

not hesitate to

shall

continue to use these functions as examples. Let / be a function defined in a neighborhood of a point x

A/ Ax

f(x

+h)- f(x

x

called the difference quotient

is

Hm

—

f{x)

)

h

+

/(xp

h)

f(x

off at x

-

/(x

)

(Ax

)

exists, it is called the derivative

of/ at x and

or sometimes, recalling the

x is

Df(x

),

common

/(x)-/(x

= Hm x->x

f'(x

x

-

x

=

h

ratio

^

0),

If

.

h

ft-0

=

x

The

.

)

—x

written

)

notation y =/(x),

dy

dx where the point x

is

indicated by (dy/dx) Xo or

is

implied by the setting of the

problem. If

f\x)

exists for all points

(an interval perhaps),

x

in

called the derived function or the derivative

If/has a derivative

at a point

x

a derivative at every point of a set D,

/'

is

continuous on a

set

it is

x

,

it

has

on D.

If

there.

the one-sided derivatives

may

They can be defined by

m

hm

,-

(1)

/(Xp

+

h)

-

/(X

)

hm

,

n

/i-o+

and

f(x

at a point

x

equality of these

is

The function f(x) =

\x\

/( X

) ,

n a necessary and sufficient condi-

two

limits.

analogous to the proof of Theorem 2.4b, and

derivatives at the origin,

+ h)-

ft-o-

The existence of a derivative tion for the existence is

said to be differentiate

D, we say that /is continuously differentiable

If the derivative fails to exist at a point still exist.

off

said to be differentiate there. If

it is

,

the domain D of f we denote by/'. It is

some subset D' of

defines there a function that

it

The proof of this

is left

as

an

result

exercise.

can easily be shown to have different one-sided

and hence /(x)

If the one-sided derivatives given

by

=

|x| is

not differentiable at x

(1) fail to exist,

=

0.

then a weaker type

may. These are given by

m (2)

,.

hm

h->o+

/(x

+ h)-

f(x

+

0) ,

n

The function sgn x has

hm /i-o-

f(x

+ h)-

f(x

-

0) .

n

derivatives in these senses at the origin,

and the

The Derivative. Chain Rule values of these are equal, but (1).

Furthermore,

There

is

it

85

does not have derivatives there in the sense of

this function is

not differentiable at the origin.

no standard notation

When we

for these one-sided derivatives.

have occasion to use a one-sided derivative, we

which we

state explicitly

mean.

An

important property of differentiable functions

Theorem.

3.9a

x

continuous at

Proof Let

If a function

A =f'(x

),

n -»


S P .{f).

| if

P

and P' are any

p

104

Integration

Proof. This is very easy to prove. Let P" be the common refinement of P and P' formed by taking as partition points all points that occur in P and P'.

Then ijp

The

first

Lemma Our

and

^

^

jjp»

last inequalities are

by

jj

p"

^

r

±t

Lemma

•

4.2b and the middle one by

4.2a.

|

as a lower

now

of lemmas has

series

bounded function

established the following facts.

For any

of numbers formed by the upper sums has

/, the class

bound any lower sum, and conversely

the class of lower

sums has

any upper sum as an upper bound. Thus

infSp^Sy for any partition P'.

Thus

inf

SP

an upper bound for the lower sums, so

is

that inf

^

SP

sup

S P >.

Now inf S P (f) is called the upper Riemann-Darboux integral of / over /: {a < x ^ b}, and sup S P >(f) the lower Riemann-Darboux integral off over /.

They

are written b

inf

S P (f)=~j f(x)dx, a

sup

We

refer to

them simply

SPif) = fj(x)dx.

as the upper and lower integrals.

We

have therefore

proved the following lemma. 4.2d

Lemma.

Iff is a bounded function on ~j

h

/,

b

J(x)dx>jJ(x)dx.

4.2e

Lemma.

Iff is bounded on

and lower bounds for f on

/,

I:

{a^ x
0,

may

P by

or

may

not have a

introducing a partition

induces also partitions of

right

+ SI > J/(x) dx + J'/(x) dx. form a lower bound for

all

the upper

sums on

b

b

Now,

it

Thus

dx > JV(x) dx + ] ]J(x)

(1)

S ab and S* will represent, {a ^ x ^ b} and one over

as with integrals, so that

there are partitions of {a

f(x) dx.

^x^

c}

and

{c

^x^

b} for

which

s«

0,

S

J-e

£p M

k

A, x

- e.

we then have

(5),

0>SP -s> £/(*) dx -

Sp(f,

£

or

J

>

f(x) dx

—

for every

2fi

£.

Thus J

By

a similar argument,

we

> J*/W

Jjc.

get

/

Since the upper integral by definition

is

b

~\

not

the proof.

J(x) less

dx.

than the lower,

this

completes |

Example

Compute x dx. Solution.

The integrand

of the integral

is

clear

4.3e, the integral

is

by

is

both monotone and continuous. The existence

Theorem 4.3a or 4.3b. And since, by Theorem of the Riemann sums, we take a sequence of

either

the limit

The Riemann Integral

simple partitions the upper sum,

Pn

{0, a/n, 2a/n,

:

as

||P||

.

.

,

-» 0, n

(a\

[ka\

- oo

and

a

x dx

it

should happen that

/ is

2

"

a

,

2

n

2

^x
n, and hence = + h where h >

that e"

Write e 2.

1

complete the proof of Theorem

Evaluate the following

limits.

some

In

cases

— x

lim x-0

: 3

lim : »-o log

\-e x%

2

.

X

-.

lim (e*

(d)

-,yr^

2

we

Estimating exactly as before,

get

/2^^Acos0^ J2k -

k

A0

:

@E* -

V

k

2/c

^_ t

:

^

Ask->0, d -—cos d9

n

= — 1 = —sin-. .

2

n/2

Thus

— dd

sin 9

=

cos

0,

—- cos

= —sin

{-oo
)]

d,0)

146

The Elementary Transcendental Functions

P

from

(both 6 and

is

nates of

P

are (cos

are indicated as positive in the figure).

and of

0, sin 9)

rotate the three points clockwise

P'[cos

Then Thus

(

— 6),

the

Q

The

+ ).

P' to Q'.

2

the lower limit taken to be zero?

gives the principal value of the arc cos x. Obtain

Show

)

fe-e\=-cosfe+o\ = smO.

3.

5.

147

x and cos x are

x

x

infinitely differentiate.

formula for arc cos

x.

CHAPTER

6

Properties

of Differentiable Functions

The Cauchy

6.1

In this chapter

Mean

Value Theorem

we obtain some further properties of differentiable functions,

most of which are consequences of the mean value theorem

(3.10c).

We

Cauchy

begin by establishing an extension of this theorem

mean

{a

^

there

is

Suppose that

Theorem.

x

^

b)

and

a point p

in

addition, g(b)

f and g

differentiable on the

are continuous on a closed interval

open interval J: {a