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 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.