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Is a Lipschitz function differentiable?

I have been wondering whether or not this property applies to all functions.

I do not need a formal proof, just the concept behind it.
Let $f: [a,b] \to [c,d]$ be a continuous function (What is more - it is uniformly continuous!) And let's assusme that it's also Lipschitz continuous on this interval.

Does this set of assumptions imply that $f$ is differentiable on $(a,b)$?

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3 Answers

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It is not always true indeed, good counterexample could be $x\mapsto |x-a|$. But rather, we have

Theorem: Radamacher theorem says every Lipschitz function is almost everywhere differentiable

Fine a nice proof of this theorem here: An Elementary Proof of Rademacher's Theorem - James Murphy or Here using distribution theory

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The function $$x \mapsto \left|x\right|$$ is Lipschitz-continuous (with $k=1$) but not differentiable at $0$.

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No, it does not imply $f$ is differentiable.

Try $f(x) = |x|$ as an example!

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