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Is it true that every Lipschitz continuous function has a continuous derivative ?

I am asking out of curiosity because I can't find a counter example. Thanks!!

EDIT : The function is differentiable

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    what is the domain and co-domain of the function? – 2017-01-21

1 Answers 1

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No. Lipschitz functions need not be differentiable at all. Try $$ f(x)=|x| $$ with Lipschitz constant 1.

It is true that Lipschitz functions are differentiable everywhere but a set of measure zero, thanks to Rademacher's Theorem.

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    what about lipschitz functions that are differentiable? – 2017-01-21
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    @JorgeFernándezHidalgo that's a good question, I am not sure of the answer. The derivative is bounded obviously, but I don't see an easy way to bound differences in the derivative range in terms of distances in the domain. Is it false? If not, any hints? – 2017-01-21
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    Thanks for the answer but this counter example is somehow artificial. I meant a function which is uniquely defined everywhere in R . – 2017-01-22
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    I don't know what you mean. This function is defined on all of R – 2017-01-22
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    sorry I meant the function is differentiable. – 2017-01-24