Counterexamples in Analysis

Counterexamples in Analysis

Counterexamples in Analysis

Bernard R. Gelbaum

University of California, Irvine

John M. H. Olmsted

Southern Illinois University

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1964 Holden-Day, Inc.

Copyright © Renewed 1992 by Bernard R. Gelbaum and John M.H. Olmsted

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2003, is an unabridged, slightly corrected republication of the 1965 second printing of the work originally published in 1964 by Holden-Day, Inc., San Francisco.

Library of Congress Cataloging-in-Publication Data

Gelbaum, Bernard R.

Counterexamples in analysis / Bernard R. Gelbaum, John M.H. Olmsted.

p. cm.

… an unabridged, slightly corrected republication of the 1965 second printing of the work originally published in 1964 by Holden-Day, San Francisco [in the Mathesis series]--T.p. verso.

Includes bibliographical references and index.

ISBN-13: 978-0-486-42875-8

ISBN-10: 0-486-42875-3

1. Mathematical analysis. I. Olmsted, John Meigs Hubbell, 1911-II. Title.

QA300.G4 2003

515–dc21

2002041784

Manufactured in the United States by Courier Corporation

42875307

www.doverpublications.com

Preface

The principal type of question asked in mathematics is, "Is statement S true?" where the statement S is of the form "Every member of the class A is a member of the class ." To demonstrate that such a statement is true means to formulate a proof of the inclusion . To demonstrate that such a statement is false means to find a member of A that is not a member of B, in other words a counterexample. To illustrate, if the statement S is Every continuous function is somewhere differentiable, then the sets A and B consist of all continuous functions and all functions that are somewhere differentiable, respectively; Weierstrass’s celebrated example of a function f that is continuous but nowhere differentiable (cf. Example 8, Chapter 3) is a counterexample to the inclusion , since f is a member of A that is not a member of B. At the risk of oversimplification, we might say that (aside from definitions, statements, and hard work) mathematics consists of two classes—proofs and counter-examples, and that mathematical discovery is directed toward two major goals—the formulation of proofs and the construction of counterexamples. Most mathematical books concentrate on the first class, the body of proofs of true statements. In the present volume we address ourselves to the second class of mathematical objects, the counterexamples for false statements.

Generally speaking, examples in mathematics are of two types, illustrative examples and counterexamples, that is, examples to show why something makes sense and examples to show why something does not make sense. It might be claimed that any example is a counterexample to something, namely, the statement that such an example is impossible. We do not wish to grant such universal interpretation to the term counterexample, but we do suggest that its meaning be sufficiently broad to include any example whose role is not that of illustrating a true theorem. For instance, a polynomial as an example of a continuous function is not a counterexample, but a polynomial as an example of a function that fails to be bounded or of a function that fails to be periodic is a counterexample. Similarly, the class of all monotonic functions on a bounded closed interval as a class of integrable functions is not a counterexample, but this same class as an example of a function space that is not a vector space is a counterexample.

The audience for whom this book is intended is broad and varied. Much of the material is suitable for students who have not yet completed a first course in calculus, and for teachers who may wish to make use of examples to show to what extent things may go wrong in calculus. More advanced students of analysis will discover nuances that are usually by-passed in standard courses. Graduate students preparing for their degree examinations will be able to add to their store of important examples delimiting the range of the theorems they have learned. We hope that even mature experts will find some of the reading new and worthwhile.

The counterexamples presented herein are limited almost entirely to the part of analysis known as real variables, starting at the level of calculus, although a few examples from metric and topological spaces, and some using complex numbers, are included. We make no claim to completeness. Indeed, it is likely that many readers will find some of their favorite examples missing from this collection, which we confess is made up of our favorites. Some omissions are deliberate, either because of space or because of favoritism. Other omissions will undoubtedly be deeply regretted when they are called to our attention.

This book is meant primarily for browsing, although it should be a useful supplement to several types of standard courses. If a reader finds parts hard going, he should skip around and pick up something new and stimulating elsewhere in the book. An attempt has been made to grade the contents according to difficulty or sophistication within the following general categories: (i) the chapters, (ii) the topics within chapters, and (iii) the examples within topics. Some knowledge of related material is assumed on the part of the reader, and therefore only a minimum of exposition is provided. Each chapter is begun with an introduction that fixes notation, terminology, and definitions, and gives statements of some of the more important relevant theorems. A substantial bibliography is included in the back of the book, and frequent reference is made to the articles and books listed there. These references are designed both to guide the reader in finding further information on various subjects, and to give proper credits and source citations. If due recognition for the authorship of any counterexample is lacking, we extend our apology. Any such omission is unintentional.

Finally, we hope that the readers of this book will find both enjoyment and stimulation from this collection, as we have. It has been our experience that a mathematical question resolved by a counterexample has the pungency of good drama. Many of the most elegant and artistic contributions to mathematics belong to this genre.

B.R.G.

Irvine, California

J.M.H.O.

Carbondale, Illinois

Table of Contents

Part I. Functions of a Real Variable

1.The Real Number System

Introduction

1.An infinite field that cannot be ordered

2.A field that is an ordered field in two distinct ways

3.An ordered field that is not complete

4.A non-Archimedean ordered field

5.An ordered field that cannot be completed

6.An ordered field where the rational numbers are not dense

7.An ordered field that is Cauchy-complete but not complete

8.An integral domain without unique factorization

9.Two numbers without a greatest common divisor

10.A fraction that cannot be reduced to lowest terms uniquely

11.Functions continuous on a closed interval and failing to have familiar properties in case the number system is not complete

a.A function continuous on a closed interval and not bounded there (and therefore, since the interval is bounded, not uniformly continuous there)

b.A function continuous and bounded on a closed interval but not uniformly continuous there

c.A function uniformly continuous (and therefore bounded) on a closed interval and not possessing a maximum value there

d.A function continuous on a closed interval and failing to have the intermediate value property

e.A nonconstant differentiable function whose derivative vanishes identically over a closed interval

f.A differentiable function for which Rolle’s theorem (and therefore the law of the mean) fails

g.A monotonic uniformly continuous nonconstant function having the intermediate value property, and whose derivative is identically 0 on an interval

2.Functions and Limits

Introduction

1.A nowhere continuous function whose absolute value is everywhere continuous

2.A function continuous at one point only (cf. Example 22)

3.For an arbitrary noncompact set, a continuous and unbounded function having the set as domain

4.For an arbitrary noncompact set, an unbounded and locally bounded function having the set as domain

5.A function that is everywhere finite and everywhere locally unbounded

6.For an arbitrary noncompact set, a continuous and bounded function having the set as domain and assuming no extreme values

7.A bounded function having no relative extrema on a compact domain

8.A bounded function that is nowhere semicontinuous

9.A nonconstant periodic function without a smallest positive period

10.An irrational function

11.A transcendental function

12.Functions y = f(u), , and u = g(x), , whose composite function y = f(g(x)) is everywhere continuous, and such that

13.Two uniformly continuous functions whose product is not uniformly continuous

14.A function continuous and one-to-one on an interval and whose inverse is not continuous

15.A function continuous at every irrational point and discontinuous at every rational point

16.A semicontinuous function with a dense set of points of discontinuity

17.A function with a dense set of points of discontinuity every one of which is removable

18.A monotonic function whose points of discontinuity form an arbitrary countable (possibly dense) set

19.A function with a dense set of points of continuity, and a dense set of points of discontinuity no one of which is removable

20.A one-to-one correspondence between two intervals that is nowhere monotonic

21.A continuous function that is nowhere monotonic

22.A function whose points of discontinuity form an arbitrary given closed set

23.A function whose points of discontinuity form an arbitrary given Fσ set (cf. Example 8, Chapter 4, and Examples 8, 10, and 22, Chapter 8)

24.A function that is not the limit of any sequence of continuous functions (cf. Example 10, Chapter 4)

25.A function with domain [0, 1] whose range for every nondegenerate subinterval of [0, 1] is [0, 1] (cf. Example 27, Chapter 8)

26.A discontinuous linear function

27.For each , n(2n + 1) functions , n, i = 1, 2, · · · , 2n + 1, satisfying:

(a)All are continuous on [0, 1]

(b)For any function f(x1, x2, · · · , xn) continuous for 0 x1, x2, · · · , xn 1, there are 2n + 1 functions , i = 1, 2, · · · , 2n + 1, each continuous on , such that

3.Differentiation

Introduction

1.A function that is not a derivative

2.A differentiable function with a discontinuous derivative

3.A discontinuous function having everywhere a derivative (not necessarily finite)

4.A differentiable function having an extreme value at a point where the derivative does not make a simple change in sign

5.A differentiable function whose derivative is positive at a point but which is not monotonic in any neighborhood of the point

6.A function whose derivative is finite but unbounded on a closed interval

7.A function whose derivative exists and is bounded but possesses no (absolute) extreme values on a closed interval

8.A function that is everywhere continuous and nowhere differentiable

9.A differentiable function for which the law of the mean fails

10.An infinitely differentiable function of x that is positive for positive x and vanishes for negative x 40

11.An infinitely differentiable function that is positive in the unit interval and vanishes outside

12.An infinitely differentiable bridging function, equal to 1 on [1, + ∞), equal to 0 on (− ∞, 0], and strictly monotonic on [0, 1]

13.An infinitely differentiable monotonic function f such that

4.Riemann Integration

Introduction

1.A function defined and bounded on a closed interval but not Riemann-integrable there

2.A Riemann-integrable function without a primitive

3.A Riemann-integrable function without a primitive on any interval

4.A function possessing a primitive on a closed interval but failing to be Riemann-integrable there (cf. Example 35, Chapter 8)

5.A Riemann-integrable function with a dense set of points of discontinuity

6.A function f such that is everywhere differentiable with a derivative different from f(x) on a dense set

7.Two distinct semicontinuous functions at a zero distance

8.A Riemann-integrable function with an arbitrary Fσ set of measure zero as its set of points of discontinuity (cf. Example 22, Chapter 8)

9.A Riemann-integrable function of a Riemann-integrable function that is not Riemann-integrable (cf. Example 34, Chapter 8)

10.A bounded monotonic limit of Riemann-integrable functions that is not Riemann-integrable (cf. Example 33, Chapter 8)

11.A divergent improper integral that possesses a finite Cauchy principal value

12.A convergent improper integral on [1, + ∞) whose integrand is positive, continuous, and does not approach zero at infinity

13.A convergent improper integral on [0, + ∞) whose integrand is unbounded in every interval of the form [a, + ∞), where a > 0

14.Functions f and g such that f is Riemann-Stieltjes integrable with respect to g on both [a, b] and [b, c], but not on [a, c]

5.Sequences

Introduction

1.Bounded divergent sequences

2.For an arbitrary closed set, a sequence whose set of limit points is that set

3.A divergent sequence {an} for which for every positive integer p

4.For an arbitrary strictly increasing sequence of positive integers, a divergent sequence {an} such that

5.Sequences {an} and {bn} such that

6.Sequences {a1n}, {a2n}, · · · such that

7.Two uniformly convergent sequences of functions the sequence of whose products does not converge uniformly

8.A divergent sequence of sets

9.A sequence {An} of sets that converges to the empty set but whose cardinal numbers → + ∞

6.Infinite Series

Introduction

1.A divergent series whose general term approaches zero

2.A convergent series and a divergent series such that an bn, n = 1, 2, · · ·

3.A convergent series and a divergent series such that an bn , n = 1, 2, · · ·

4.For an arbitrary given positive series, either a dominated divergent series or a dominating convergent series

5.A convergent series with a divergent rearrangement

6.For an arbitrary conditionally convergent series and an arbitrary real number x, a sequence , where for n = 1, 2, · · · , such that

7.Divergent series satisfying any two of the three conditions of the standard alternating series theorem

8.A divergent series whose general term approaches zero and which, with a suitable introduction of parentheses, becomes convergent to an arbitrary sum

9.For a given positive sequence {bn} with limit inferior zero, a positive divergent series whose general term approaches zero and such that

10.For a given positive sequence {bn} with limit inferior zero, a positive convergent series such that

11.For a positive sequence {cn} with limit inferior zero, a positive convergent series and a positive divergent series such that an/bn = cn, n = 1, 2, · · ·

12.A function positive and continuous for x 1 and such that

13.A function positive and continuous for x 1 and such that

14.Series for which the ratio test fails

15.Series for which the root test fails

16.Series for which the root test succeeds and the ratio test fails

17.Two convergent series whose Cauchy product series diverges

18.Two divergent series whose Cauchy product series converges absolutely

19.For a given sequence , n = 1, 2, · · · of positive convergent series, a positive convergent series that does not compare favorably with any series of

20.A Toeplitz matrix T and a divergent sequence that is transformed by T into a convergent sequence

21.For a given Toeplitz matrix T = (tij), a sequence {aj} where for each j, aj = ±1, such that the transform {bi} of {aj} by T diverges

22.A power series convergent at only one point (cf. Example 24)

23.A function whose Maclaurin series converges everywhere but represents the function at only one point

24.A function whose Maclaurin series converges at only one point

25.A convergent trigonometric series that is not a Fourier series

26.An infinitely differentiable function f(x) such that and that is not the Fourier transform of any Lebesgue-integrable function

27.For an arbitrary countable set , a continuous function whose Fourier series diverges at each point of E and converges at each point of [−π, π]\E

28.A (Lebesgue-) integrable function on [−π, π] whose Fourier series diverges everywhere

29.A sequence {an} of rational numbers such that for every function f continuous on [0, 1] and vanishing at 0 (f(0) = 0) there exists a strictly increasing sequence {nν} of positive integers such that, with n0 ≡ 0: the convergence being uniform on [0, 1]

7.Uniform Convergence

Introduction

1.A sequence of everywhere discontinuous functions converging uniformly to an everywhere continuous function

2.A sequence of infinitely differentiable functions converging uniformly to zero, the sequence of whose derivatives diverges everywhere

3.A nonuniform limit of bounded functions that is not bounded

4.A nonuniform limit of continuous functions that is not continuous

5.A nonuniform limit of Riemann-integrable functions that is not Riemann-integrable (cf. Example 33, Chapter 8)

6.A sequence of functions for which the limit of the integrals is not equal to the integral of the limit

7.A sequence