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.
Best Answer
There are several, at least on a bounded interval, you have
Differentiability $\Rightarrow$ Lipschitz continuity $\Rightarrow$ Hölder continuity (with decreasing exponent) $\Rightarrow$ Absolute continuity $\Rightarrow$ continuity
which are some of the ones that are used more often.