This really is the result of Heine-Cantor. Try using the Lebesgue Covering Lemma

EDIT: This is probably overkill, but I don't have time to think about it right now. There's absolutely an easier way that the one I mentioned above (maybe) if you're given that a continuous function on closed intervals is unif. cont. Let me think about it.

I prove the much more general result I alluded to earlier here (last theorem).