f differentiable implies f' continuous.

Suppose that f is differentiable on (a, b) and that f ' is increasing on (a, b). Prove that f ' is continuous on (a, b).

I'm actually at a lost on how to do this problem. The method I keep finding myself trying is to assume that some point on f ' is not continuous. Then show this a contradiction. The problem is I keep finding myself assuming that the rest of f ' is continuous and I have no clue how f ' increasing contributes to this proof.

It seems to me that any differentiable function would have f ' has continuous, otherwise, the original function would have a 'sharp corner' causing it to not be differentiable.

Thanks in advance!