Counterexamples on Uniform Convergence

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

Este título pertence ao(s) assunto(s) indicados(s). Para ver outros títulos clique no assunto desejado.

• Cálculo e análise matemática

counterexamples, mathematical analysis, calculus, proper and improper integrals, real analysis, uniform convergence