An Introduction to Real Analysis

R. Agarwal, C. Flaut and D. O’Regan’s An Introduction to Real Analysis is meant for upper-level undergraduate mathematics students. The book is divided into thirty short chapters, none of them more than eleven pages long. All chapters include exercises and hints. The short chapters are nicely written, with very clear proofs and 122 examples, making the book easy to read and work with.

The topics are the expected ones in a textbook of this type (see the table of contents for specifics). One unusual topic is Froda’s theorem describing the set of points of discontinuities of a monotone function. Convex functions are introduced in chapter 17. Metric spaces are introduced only at the end.

Real analysis is a sea full of mathematical jewels. Mathematicians have to choose among them to include in their introductory books. A book shines in as much as the chosen jewels shine. In this book, they are many of them: the equivalence between the Dedekind property and the completeness axiom of the real line, Froda’s theorem, Darboux’s theorem, the Riemann-Lebesgue theorem, and very many others.

The book contains 289 exercises. Most of them should be considered as an integral part of the text. Readers and student are advised to do all of them, not only for their beauty but because they are very interesting results as well.

I would have liked to see more than just the three figures. I believe that figures and graphs make it easier for a student to grasp certain notions and to have a good feeling of the subject.

This book is one of the best books I have seen on real analysis. I recommend it to any student who would really understand real analysis.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.