Thread: 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!

You need to use the fact that f' is increasing - the statement is not true otherwise. The proof probably would make use of the fact that derivative attains intermediate values.

The derivative of a function is not necessarily continuous but it does satisfy the "intermediate value" property: if f is differentiable on [a, b] and c is a number between f'(a) and f'(b), then there exist an such that is between f'(a) and f'(b). A consequence of that is that if and exist, then they are equal and the derivative is continuous there. So if f' is NOT continuous at least one of those one-sided limits does not exist. Use "increasing" to show that cannot happen.

As suggested satisfies the intermediate value property now for any arbitrary , given an consider by intermediate value property such that now lets choose we always have . Hence is continuous on (a,b).