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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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marked as duplicate by Isaac Solomon, Américo Tavares, froggie, Norbert, Nate Eldredge Nov 22 '12 at 18:17

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Yes. The standard example is the Weierstrass function. – froggie Nov 22 '12 at 17:42
Or a Wiener process - note the "fractal" nature of the image. – kahen Nov 22 '12 at 17:44
Yes, most (in the sense of category) continuous functions are nowhere differentiable. – André Nicolas Nov 22 '12 at 17:49
this might help – Norbert Nov 22 '12 at 17:56
@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. – André Nicolas Nov 22 '12 at 18:10
up vote 1 down vote accepted

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 $C[0,1]$. – kahen Nov 22 '12 at 18:04

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