Counterexamples on Uniform Convergence

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.

"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 convergencebased 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 toall exercises, which is available via a companion website" Mathematical Reviews, Sept 2017

• Preface ixList of Examples xiList of Figures xxixAbout the Companion Website xxxiiiIntroduction xxxvI.1 Comments xxxvI.1.1 On the Structure of This Book xxxvI.1.2 On Mathematical Language and Notation xxxviiI.2 Background (Elements of Theory) xxxviiiI.2.1 Sequences of Functions xxxviiiI.2.2 Series of Functions xliI.2.3 Families of Functions xliv1 Conditions of Uniform Convergence 11.1 Pointwise, Absolute, and Uniform Convergence. Convergence on a Set and Subset 11.2 Uniform Convergence of Sequences and Series of Squares and Products 151.3 Dirichlet’s and Abel’s Theorems 31Exercises 39Further Reading 422 Properties of the Limit Function: Boundedness, Limits, Continuity 452.1 Convergence and Boundedness 452.2 Limits and Continuity of Limit Functions 512.3 Conditions of Uniform Convergence. Dini’s Theorem 682.4 Convergence and Uniform Continuity 79Exercises 88Further Reading 933 Properties of the Limit Function: Differentiability and Integrability 953.1 Differentiability of the Limit Function 953.2 Integrability of the Limit Function 117Exercises 128Further Reading 1314 Integrals Depending on a Parameter 1334.1 Existence of the Limit and Continuity 1334.2 Differentiability 1444.3 Integrability 154Exercises 162Further Reading 1665 Improper Integrals Depending on a Parameter 1675.1 Pointwise, Absolute, and Uniform Convergence 1675.2 Convergence of the Sum and Product 1765.3 Dirichlet’s and Abel’s Theorems 1855.4 Existence of the Limit and Continuity 1925.5 Differentiability 1985.6 Integrability 202Exercises 210Further Reading 214Bibliography 215Index 217