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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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    Yes. The standard example is the [Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function).2012-11-22
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    Or a [Wiener process](https://en.wikipedia.org/wiki/Wiener_process#Brownian_scaling) - note the "fractal" nature of the image.2012-11-22
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    Yes, most (in the sense of category) continuous functions are nowhere differentiable.2012-11-22
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    @AndréNicolas Could you elaborate?2012-11-22
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    [this](http://math.stackexchange.com/questions/136445/holder-continuous-but-not-differentiable-function/136500) might help2012-11-22
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    @Andris: Look at the continuous functions from $[0,1]$ to the reals, under the sup norm. Then the set of such functions that are continuous *somewhere* is meager. Standard result, I think it is in Oxtoby, but it should not be hard to track down a proof on the Web.2012-11-22
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    @André: your second "continuous" should be a "differentiable".2012-11-22
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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