Problems in Real Analysis: Advanced Calculus on the Real Axis

The book under review, as its title shows, essentially is a problem book in real analysis, chosen mostly from mathematical Olympiads and from problem journals. It also contains some nice counterexamples and proofs of some well-known theorems.

Every chapter of the book starts with a review of basic concepts and key results, often with no proof and no additional explanation. Then begin the problems with solutions. They are tricky elementary problems, and the expanded solutions teach the reader how to solve such problems. Each chapter concludes with a number of further problems.

The book focuses on analysis on the real line, which is also known as advanced real calculus. It covers the normal topics of elementary analysis. In particular, there are good accounts of infinite series and products and of inequalities, both of which are important topics for anyone doing mathematics. The book provides a very good and rather complete collection of the main results and tools in analysis and advanced calculus, which is fine for the people looking for tools to apply and do not need detailed proofs, such as students preparing for Olympiads or engineers.

Since the book doesn’t cover analysis concepts on abstract spaces, it is not usable as a text in an undergraduate course that does analysis on metric spaces. But, problems and counterexamples are very useful in any course in Mathematical Analysis, so the book can be used as a supplement. Moreover, some of them seem very fine for courses in calculus and even, if some cases, for high school classes.

The book includes some brief and useful historical notes, but the reader should (as always) be ready to check them; I think it is necessary here to mention some points:

• A well-known function is introduced as the Riemann function at the page 152 (and also page 327), and then as the Dirichlet function at the page 198.
• On page 186 we read “In fact, the first example of a continuous function that is nowhere differentiable is due to Weierstrass (1872), who shocked the mathematical world by showing that …” In actual fact, the first example of such functions is due to Bernard Bolzano around 1830 (but it was only published in 1922, after being rediscovered a few years earlier).  

Moreover, the book contains a number of typos, some of which are:

• The graphs of the functions in Figures 5.2 and 5.3 (illustrating the concepts of “cusp point” and “vertical tangent”) need to be exchanged.
• On page 263, the first example for the convex functions is not introduced completely.
• At the end of page 314 we read “We end this preliminary section by listing a number of useful integration formulas.” But no list follows.

Typos aside, however, the book under review is a collection of interesting and fresh problems with detailed solutions. The target audience seems to be students preparing for Olympiads and other competitions, but undergraduate students, mathematics teachers and professors of Mathematical Analysis and Calculus courses may also find interesting things here.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.