Suppose that f : (a, b) → R is continuous and that f(r) = 0 for ever rational number r є (a, b). Prove that f(x) = 0 for all x є (a, b).
Very nice- and not at all what I was thinking of. I was thinking of a proof by contradiction. Suppose there exist some a such that . Let . Then for all , there exist rational x in the interval . For that x, but , contradicting the fact that f is continuous.