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

## Books

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 the book is threefold. First, the authors provide a brief survey and discussion of principal results of the theory of uniform convergence in real analysis. Second, the book 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.

The features of the book include:

• 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. The book is also a valuable resource 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. Andrei 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. Ludmila Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

Preface ix

List of Examples xi

List of Figures xxix

About the Companion Website xxxiii

Introduction xxxv

I.1 Comments xxxv

I.1.1 On the Structure of This Book xxxv

I.1.2 On Mathematical Language and Notation xxxvii

I.2 Background (Elements of Theory) xxxviii

I.2.1 Sequences of Functions xxxviii

I.2.2 Series of Functions xli

I.2.3 Families of Functions xliv

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 15

1.3 Dirichlet’s and Abel’s Theorems 31

Exercises 39

Further Reading 42

2 Properties of the Limit Function: Boundedness, Limits, Continuity 45

2.1 Convergence and Boundedness 45

2.2 Limits and Continuity of Limit Functions 51

2.3 Conditions of Uniform Convergence. Dini’s Theorem 68

2.4 Convergence and Uniform Continuity 79

Exercises 88

Further Reading 93

3 Properties of the Limit Function: Differentiability and Integrability 95

3.1 Differentiability of the Limit Function 95

3.2 Integrability of the Limit Function 117

Exercises 128

Further Reading 131

4 Integrals Depending on a Parameter 133

4.1 Existence of the Limit and Continuity 133

4.2 Differentiability 144

4.3 Integrability 154

Exercises 162

Further Reading 166

5 Improper Integrals Depending on a Parameter 167

5.1 Pointwise, Absolute, and Uniform Convergence 167

5.2 Convergence of the Sum and Product 176

5.3 Dirichlet’s and Abel’s Theorems 185

5.4 Existence of the Limit and Continuity 192

5.5 Differentiability 198

5.6 Integrability 202

Exercises 210

Further Reading 214

Bibliography 215

Index 217

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