This can be thought of either as a brief introduction to real analysis, or as a rigorous calculus book: it proves nearly all the facts that are used in single- and multi-variable calculus, but generally does not go beyond that to the more general problems considered in real analysis. This is a 1986 Dover unaltered reprint of the 1968 edition from Scott, Foresman.
The book does cover some topics that are usually not touched on in calculus, such as some theory of complete metric spaces, uniform convergence and uniform continuity, and the inverse and implicit function theorems. Unlike most calculus courses, everything is done analytically rather than geometrically; for example, the transcendental functions are defined by power series. The book does not construct the real numbers but develops them axiomatically. There is an introductory chapter on set theory, but set theory is used in the book primarily for notation rather than as a foundation.
The exercises are excellent; there are lots of them, they cover many important additional results, and they have a wide range of difficulty. Usually each problem set starts out with exercises to find some examples of something, and then switches over to proving things. There are no answers or hints given for the exercises, and there aren’t many worked examples, so the audience is probably upper-division undergraduates.
A more recent book that is similar in many ways is Ross’s Elementary Analysis: The Theory of Calculus. Ross sticks more closely to the single-variable calculus syllabus and has few excursions into more advanced topics and nothing on multi-variable calculus. Ross’s book is probably better for present-day students, because it has many more worked examples, and it has (brief) answers to many of the exercises in the back of the book. Both are very good books, but differ in scope and in the amount of hand-holding given. Both books are much more elementary than traditional real-analysis texts such as Rudin’s Principles of Real Analysis and Apostol’s Mathematical Analysis.
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.