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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.

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Probably not what you are looking for: Real functions which are convex on an interval are continuous on that interval. They are also differentiable almost everywhere though. – Tim Seguine Jun 14 '14 at 11:04
up vote 13 down vote accepted

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.

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A function $f$ is called Lipschitz continuous if there is a (uniform) constant $K$ such that

$$|f(x) - f(y)| \le K |x - y|$$

for all $x, y$ in the domain of $f$ (alternatively, you can define a function to be locally Lipschitz if for each compact subset of the domain such a constant exists). Lipschitz continuity implies continuity (and in fact uniform continuity), but it does not imply differentiability - as a slightly trivial example, $|x|$ is Lipschitz but not differentiable at $0$. Being Lipschitz does imply differentiability almost everywhere, though (and in fact, $f' \in L^{\infty}$).

Alternatively, you can consider absolute continuity, which again implies differentiability only almost everywhere.

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Sure, there's Lipschitz continuity.

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For one, Holder continuity. See here.

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