Rudin W. Principles of Mathematical Analysis

3rd Edition. — McGraw-Hill, Inc., 1976. — (International series in pure and applied mathematics) — ISBN: 0-07-054235-X.The present edition covers essentially the same topics as the second one, with some additions, a few minor omissions, and considerable rearrangement. The material on functions of several variables is almost completely rewritten, with many details filled in, and with more examples and more motivation. The proof of the inverse function theorem—the key item in Chapter 9—is simplified by means of the fixed point theorem about contraction mappings. Differential forms are discussed in much greater detail. Several applications of Stokes’ theorem are included.
As regards other changes, the chapter on the Riemann-Stieltjes integral has been trimmed a bit, a short do-it-yourself section on the gamma function has been added to Chapter 8, and there is a large number of new exercises, most of them with fairly detailed hints.The Real and Complex Number Systems
Basic Topology
Numerical Sequences and Series
Continuity
Differentiation
The Riemann-Stieltjes Integral
Sequences and Series of Functions.
Some Special Functions
Functions of Several Variables
Integration of Differential Forms
Tbe Lebesgue Tbeory