Given a function f: [0,1] \to \mathbb{R}. Suppose f(x) = 0 if x is irrational and f(x) = 1/q if x = p/q, where p and q are relatively prime.
Prove that f is Riemann integrable.
Let f be a real uniformly continuous function on the bounded
set A in \mathbb{R}^1. Prove that f is bounded
on A.
Since f is uniformly continuous, take \epsilon = m, \exists \delta > 0
such that
|f(x)-f(p)| < \epsilon
whenever |x-p|<\delta and x,p \in A
Now we have
|f(x)| < m +...