This is a concise, very clearly-written undergraduate textbook in classical analysis, that includes a very broad selection of the most important theorems in the subject. The content and approach are severely classical, and in fact this book could have been written seventy-five years ago. (I am not complaining. No one did write it seventy-five years ago, and I am glad we have it at last.) The Preface cites the concrete topics that used to be in advanced calculus courses in the 1950s and states, “The purpose of this book is to recover the lost topics and introduce others.” The book is very much in the tradition of Hardy, Littlewood, Titchmarsh, and Landau.
The book starts with a brief development of the topology of the real line, focusing on limit theorems. After that, each chapter tends to be independent, although cross-references to other chapters are given when needed. Some topics, such as equidistribution, do pull together previous material from many areas. Each chapter ends with a wealth of problems, of moderate difficulty: not drill or straightforward applications of material in the chapter, but also noticeably easier than the results proven in the chapter. Very Good Feature: brief biographical and historical notes scattered through the text.
A very different book with similar subject matter is Hongwei Chen’s Excursions in Classical Analysis. Duren’s book is oriented toward the most important theorems of classical analysis, while Chen’s book is oriented toward particular problems (in fact much of Chen’s book investigates problems from the American Mathematical Monthly problems section and from the annual William Lowell Putnam Competition). Chen’s book uses many of the same classical techniques that are in Duren, but includes newer methods such as the use of computer algebra systems, numerical experiments, and Wilf–Zeilberger summation.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.