Real Analysis: Foundations and Functions of One Variable

By present-day American standards this is a rigorous single-variable calculus book. It is not a real analysis book in the sense of mainline texts such as Rudin’s Principles of Mathematical Analysis or Apostol’s Mathematical Analysis.

I wasn’t very happy with the approach taken here. It stays within the boundaries of calculus and doesn’t have any of the abstract properties that we associate with real analysis. It doesn’t serve as a springboard to more advanced topics. For example, there are no metric spaces or topology. There are not any sequences of functions. Even power series are omitted, except for some of the exercises. (The trigonometric functions are defined geometrically, just like they are in trigonometry, and the exponential and logarithmic functions are defined pointwise by limits of sequences.)

Within these limitations, the book does a reasonable job. The book begins with some materials on logic, proofs, and set theory. The real numbers are treated axiomatically rather than by construction. The book has thorough coverage of sequences, a little bit about infinite series, and reasonably-thorough coverage of differentiation and integration, including Stieltjes integrals. The big strength of the book is its exercises, which are very challenging, although most of them are about particular functions rather than properties of functions in general. The back of the book has both hints and solutions for many of the exercises.

I suspect that a big reason for the book’s weakness is that it is only half a book: it is a translation of the first of a two-volume set in Hungarian. I haven’t been able to find a good description of the second volume, and in any case there seems to be no plan to translate it into English.

The book seems to be positioned as a bridge course that would follow a cookbook calculus sequence but precede a rigorous real analysis course. A better choice would be Ross’s Elementary Analysis: The Theory of Calculus. It has generally the same coverage but mixes in the more abstract concepts in palatable quantities.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.