Goldberg Richard R. Methods of Real Analysis

2nd edition. — John Wiley & Sons, Inc., 1976. — 402 p.This book is intended as a one-year course for students who have completed an ordinary sequence of courses in elementary calculus. It presents in rigorous fashion basic material on the fundamental concepts and tools of analysis—functions, limits, continuity, derivatives and integrals, sequences, and series. Most of the difficult points usually glossed over in elementary courses are dealt with in detail, as well as many more advanced topics designed to give a good background for (and, hopefully, a taste of) modern analysis and topology. In particular, there are treatments of metric spaces and Lebesgue integration, topics that are often reserved for more advanced courses. Also included are many smaller but interesting topics not usually presented in courses at this level; these topics include Baire category and discontinuous functions, summability of series, the Weierstrass theorem on approximation of continuous functions by polynomials, and a proof of the standard existence theorem for differential equations from
the point of view of fixed-point theory.
The book is written at the same level as texts for traditional "advanced calculus" courses, but does not consider topics in "several variables." Material on differentials and vector calculus, in our opinion, can be understood best from the point of view of modern differential geometry and belong in a separate course.