It is a well-known fact that differentiability implies continuity. My question is this: is there some condition for a function that is both weaker than differentiability and stronger than continuity? I.e., is there a condition that "guarantees" continuity that does not also guarantee differentiability?
Edit: I realized immediately after posting this that I did not give enough thought to asking this question. The question that I meant to ask (which, I think, is more interesting) is given here.