This is an introductory text in real analysis, aimed at upper-division undergraduates. The coverage is similar to that in Rudin’s Principles of Mathematical Analysis and Apostol’s Mathematical Analysis. This book differs from these earlier books primarily in being more talkative: explanations are written out at greater length, there are more worked examples, and there is a much larger number of exercises at all levels of difficulty.
One unusual feature of the present book is that in the body of each chapter the theorems are motivated, stated in detail, and illustrated through examples — but not proven. Complete proofs are given in a lengthy appendix to each chapter. The chapters are like survey articles, with supplements. I wasn’t completely satisfied with this approach, because at this level of course we are primarily interested in the proofs, but it is workable.
Some topics are omitted. Functions of several variables are included, but not vector calculus (vector-valued functions). The integration is strictly Riemannian, although sets of measure zero are included, as are limited versions of the Lebesgue convergence theorems.
Apart from the segregation of the proofs, I had just a couple of minor gripes. The book uses the reversed bracket notation for open intervals, for example ]0, 1[ is the open interval from 0 to 1. It also uses a mixed reference style, where references cited in the body have their bibliographic information given there, while there is a separate, disjoint list of References and Suggestions for Further Study.
Although the material is classical, it is not what most people call “classical analysis,”. The book does not deal with classical topics such as inequalities, approximation of functions, and special numbers and functions (as does, for example, Duren’s recent Invitation to Classical Analysis).
Bottom line: A good choice for students who find Rudin too austere.
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.