So, a certain discussion occured in class today... If f is differentiable, is f ' continuous? At first sight, there seems no reason to think so. However, we couldn't think any counterexample. It also seems logical that f' is continuous since otherwise f wouldn't be differentiable. For example, suppose f(x) = ln x, for x > 0 Then f'(x) = 1/x. Yes, this is discontinuous, but it's not for the domain x > 0. So, the question remains: If f is differentiable, is f ' continuous?