Mathematical Analysis – Tom Apostol – 2nd Edition

In this new edition, of a more modern spirit than the first excellent one, one can repeat the praise that was made previously: its precise and rigorous style, in a program balanced but sufficiently broad, gives character of basic text.
 
It contains topics such as the system of real numbers and complex numbers; Some basic notions of set theory; Elements of topology in sets of points; Limits and continuity; Derivatives; Functions of bounded variation and rectifiable curves; The Riemann-Stieltjes integral; Infinite series and infinite products; Inheritance of functions; The Lebesgue integral; Fourier series and Fourier integrals; Differential calculation of several variables. Implicit functions and extreme problems; Multiple integrals of Riemann; Multiple Lebesgue integrals; Cauchy’s theorem and residual calculus.