# Is there a continous function which does not have a derivative in any of its points? [duplicate]

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?

-

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

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

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.
You can do even better: $A$ is co-meagre in $C[0,1]$. – kahen Nov 22 '12 at 18:04