Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals

A comprehensive and thorough analysis of concepts and results on uniform convergence

     Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals presents counterexamples to false statements typically found within the study of mathematical analysis and calculus, all of which are related to uniform convergence. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending on a parameter. The exposition is restricted to the main definitions and theorems in order to explore different versions (wrong and correct) of the fundamental concepts and results.  

     The goal of this book is threefold. First, the authors provide a brief survey and discussion of principal results of the theory of uniform convergence in real analysis. The book also aims to help readers master the presented concepts and theorems, which are traditionally challenging and are sources of misunderstanding and confusion. Finally, this book illustrates how important mathematical tools such as counterexamples can be used in different situations.

     In addition, this book features:

An overview of important concepts and theorems on uniform convergence Well–organized coverage of the majority of the topics on uniform convergence studied in analysis courses An original approach to the analysis of important results on uniform convergence based on counterexamples Additional exercises at varying levels of complexity for each topic covered in the book A supplementary  Instructor′s Solutions Manual containing complete solutions to all exercises, which is available via a companion website  

     Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals is an appropriate reference and/or supplementary reading for upper–undergraduate and graduate–level courses in mathematical analysis and advanced calculus for students majoring in mathematics, engineering, and other sciences. This book is also a valuable source for instructors teaching mathematical analysis and calculus.

Andrei Bourchtein, PhD, is Professor in the Department of Mathematics at Pelotas State University in Brazil.  The author of more than 100 referred articles and five books, his research interests include numerical analysis, computational fluid dynamics, numerical weather prediction and real analysis. Dr. Bourchtein received his PhD in Mathematics and Physics from the Hydrometeorological Center of Russia.

Ludmila Bourchtein, PhD,is Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University in Brazil.  The author of more than 80 referred articles and three books, her research interests include real and complex analysis, conformal mappings and numerical analysis.  Dr. Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

Preface ix

List of Examples xi

List of Figures xxv

Introduction xxix

0.1 Comments xxix

0.1.1 On the structure of this book xxix

0.1.2 On mathematical language and notation xxxi

0.2 Background (elements of theory) xxxii

0.2.1 Sequences of functions xxxii

0.2.2 Series of functions xxxv

0.2.3 Families of functions xxxviii

References xliii

1 Conditions of Uniform Convergence 1

1.1 Pointwise, absolute and uniform convergence. Convergence on a set and subset 1

1.2 Uniform convergence of sequences and series of squares and products 14

1.3 Dirichlet s and Abel s Theorems 29

Exercises 38

References 40

2 Properties of the Limit Function: Boundedness, Limits, Continuity 43

2.1 Convergence and boundedness 43

2.2 Limits and continuity of limit functions 48

2.3 Conditions of uniform convergence. Dini s Theorem 64

2.4 Convergence and uniform continuity 74

Exercises 84

References 88

3 Properties of the Limit Function: Differentiability and Integrability 91

3.1 Differentiability of the limit function 91

3.2 Integrability of the limit function 112

Exercises 123

References 125

4 Integrals Depending on a Parameter 127

4.1 Existence of the limit and continuity 127

4.2 Differentiability 135

4.3 Integrability 147

Exercises 155

References 158

5 Improper Integrals Depending on a Parameter 159

5.1 Pointwise, absolute and uniform convergence 159

5.2 Convergence of the sum and product 168

5.3 Dirichlet s and Abel s Theorems 176

5.4 Existence of the limit and continuity 182

5.5 Differentiability 188

5.6 Integrability 191

Exercises 198

References 201

Bibliography 203

Index 205