Problem Solving Set 13

16 July 2012

1. Find all continuous functions f : ℝ → ℝ such that f(x) − f(y) is rational for all reals x and y such that x − y is rational.

2. Is it true or false that for each real number ϵ > 0 there exist positive integers m and n such that

0 < \sqrt{n} − \sqrt{m} − π < ϵ ?