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Possible Duplicate:
Are Continuous Functions Always Differentiable?

Is there a continous function (continous in every one of its points) which is not differentiable in any of its points?

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    @TonyK: Thanks, yes, differentiable.2012-11-22

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Yes, for example the Weierstraß function.

One can actually show that the set $A:= \{f \in C[0,1]; f$ has no right-derivative in any point in $[0,1)\}$ is dense in $C[0,1]$ and uncountable.

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    You can do even better: $A$ is co-meagre in $C[0,1]$.2012-11-22