University. For x,y ∈ R, we have |f(x)− f(y)| = x− y − x 1+x2 + y 1+y2 ≤ |x− y|+ x 1+x2 − y 1+y2 . Problems in Real Analysis teaches the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in Principles of Real Analysis, Third Edition . It is also instructive for graduate students who are interested in analytic number theory. This unique book provides a collection of more than 200 mathematical problems and their detailed solutions, which contain very useful tips and skills in real analysis. 2. fis discontinuous on the rationals. Problems and Solutions Igor Yanovsky 1. Helpful? 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and f(x) = X fn:rn