Suppose f: R--> R is continuous and f(r)= r^2 for each rational number r. Determine f(2) and justify your conclusion. Solution We claim that f(x) = x2 for all real x. Clearly there is nothing to prove if x is rational. Suppose x is irrational. COnsider a sequence {rn} of rational numbers converging to x. Now since f is continuous f(rn) -> f(x). But f(rn) = rn2 for each n. ANd we know that if a sequence {an} converges to a say, then the sequence an2 converges to a2. Hence f(rn) converges to (lim rn)2 = x2. THis completes teh proof. Hence f(2) = (2)2 =2..