Real analysis is an area of mathematics that deals with sets and sequences of real numbers , as well as functions of one or more real variables . Real analysis , or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers . In particular , it deals with the analytic properties of real functions and sequences , including convergence and limits of sequences of real numbers , the calculus of the real numbers , and continuity , smoothness and related properties of real-valued functions .