A rational number r = p/q, where p, q are in Z, is said to be properly reduced if p and q (q > 0) have no common integral factor other than +1 or -1. Define the function f as follows;
f(x) = q , if x = p/q, properly reduced.
f(x) = 0 , if x is irrational.
Prove that for every real number x, f fails to be bounded at x.
Note: f: A -> B is bounded if there exist a real number M > 0 such that |f(x)| <= M for every x in A.