Arashilabar This is an awesome book that every serious math student should own, especially graduate students preparing for qualification exams. Depending on your preference, the notation can sometimes be a little awkward, but it is neither incorrect nor ambiguous; that is, it is still correct and clear, depending on your understanding. Dover Books on Mathematics Paperback: The book conterexamples some important examples, but most of them are too arcane and inelegant. There was a counterexampls filtering reviews right now. My library Help Advanced Book Search.
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File: PDF, 9. Gelbaum and John M. Olmsted All rights reserved. Bibliographical Note This Dover edition, first published in , is an unabridged, slightly correct ed republication of the second printing of the work originally published in by Holden Day, Inc. Gelbaum, John M. Includes bibliographical references and index.
ISBN 0 pbk. Mathematical analysis. Y Preface is impossible. 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. The audience for whom this book is intended is broad and varied. 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. 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 vi Preface of related material is assumed on the part of the reader, and therefore only a minimum of ex p osi ti on is provided. A substantial bibliography is included in the back of the book, and fre quent reference is made to the arti cl e s and books listed there.
These references are designed both to guide the rea der in finding fu rth er information on various subjects, and. If du e recognition for the authorship of any counterexample is lacking, we extend our apology. Any such omission is unintentional. Many of the most elegant and artistic.
Irv1:ne, California Carbondale, Illinois B. Contents Pari I. Functions of a Real Variable 1. The Real Number System Introduction 3 1. An infinite field that cannot be ordered 13 2. A field that is an ordered field in two distinct ways An ordered field that is not complete A non-Archimedean ordered field 15 5.
An ordered field that cannot be completed 16 6. An ordered field where the rational numbers are not dense 16 7. An ordered field that is Cauchy-complete but not complete 17 8. An integral domain without unique factorization 17 9. Two numbers without a greatest common divisor 18 A fraction that cannot be reduced to lowest terms uniquely 18 n.
Functions continuous on a closed interval and failing to have familiar properties in case the number system is not complete 18 a. A function continuous on a closed interval and not bounded there and therefore, since the interval is bounded, not uniformly continuous there 19 h. A function continuous and bounded on a closed interval but not uniformly continuous there 19 c.
A function uniformly continuous and therefore bounded on a closed interval and not possessing a maximum value there 19 viii , Table of Contents d. A function continuous on a closed interval and failing to have the intermediate value property A nonconstant differentiable function whose derivative vanishes identically over a closed interval f. A monotonic uniformly continuous nonconstant function having the intermediate value property, and whose derivative is identically 0 on an interval e.
Functions and Lhnits Introduction I. 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 An irrational function II. A transcendental function For a positive sequence tive convergent Series for which the root test succeeds and the ratio test fails Two convergent series whose Cauchy product series diverges Two divergent series whose Cauchy product series converges absolutely L inferior zero, a. Example 24 68 Table of Contents 6. A means of assigning an arbitrarily large finite or infinite area to the lateral surface of a right circular 8.
A plane set of arbitrarily small plane measure within which the direction of a line segment of unit length can be reversed by means of a continuous motion Metric and Topological Spaces Introduction 1.
A decreasing sequence of nonempty closed and bounded sets with empty intersection 2. An incomplete metric space with the discrete topology 3. A decreasing sequence of nonempty closed balls in plete metric space with empty intersection 4.
Open and closed balls, a 0. A topological space X and a subset Y such that the limit points of Y do not form a closed set 7. A topological space in which limits of sequences are not unique 8.
A separable space with a nonseparable subspace 9. Function Spaces Introduction 1. Two monotonic functions whose sum is not monotonic 2. Two periodic functions whose sum is not periodic 3. Two semicontinuous functions whose sum is not semicontinuous 4. Two functions whose squares are Riemann-integrable and the square of whose sum is not Riemann,.
Two functions whose squares are Lebesgue-integrable and the square of whose sum is not Lebesgue-integrable 6. A function space that is a linear space but neither an algebra nor a lattice 7.
A linear function space that is an algebra but not a lattice 8. A linear function space that is a lattice but not an algebra 9. Two metrics for the space C [O, 1] of functions continuous on [0, 1] such that the complement of the unit ball in one is dense in the unit ball of the other Bibliography Special SYllbols Index Errata xxiv Functions of a Part I Real Variable Chapter 1 The Real Number System Introduction We begin by presenting some definitions and notations that are basic to analysis and essential to this first chapter.
These will be given in abbreviated form with a minimum of explanatory discussion. For a more detailed treatment see , , , and  of the Bibliography. If A is any set of obj ects, the statement a is a member of A is written a E A.
The union and intersection of the two sets A and B can therefore be defined :. For convenience, members of sets will often be called points. The Real Number System B. For a ddi tion and Illultiplication : The distributive law holds more precisely. The member - x of ;Y, of A iii , is called the negative, or additive inverse, of x. The member 1 of ff, of B ii , is called the one, or uni ty, or Ilultiplicative identity, of ;Yo The member X- I of ;Y, of B iii , is called the reciprocal, or multiplicative inverse, of X.
Division is a "from-to" operation and not an "on into" operation since "division by zero" is excluded. In case the commutative law A iv holds, 9 is called an Abelian or cOlllmutative group. Thus, with respect to addition any field is an Abelian additive group. P Definition II.
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