Function example? Continuous everywhere, differentiable nowhere 
Possible Duplicate:
Are Continuous Functions Always Differentiable? 

If such a function exists, can anyone give an example of a function $f(x) : \mathbb{R} \longrightarrow \mathbb{R}$  that is continuous for all $x \in \mathbb{R}$ but differentiable nowhere?
 A: Another popular example is what I know as Takagi's Function.
It is somehow different from the Weierstrass Function in that it is not constructed as a uniform limit of differentiable functions. However, it is a uniform limit of continuous functions in a way that the points of non-differentiability populate the "whole interval" (if that point of view makes any sense...).
A: A very famous example - and by far the most important when it comes to practical applications (finance: option pricing!) - is the Wiener process.
A: See Wikipedia's page on the Weierstrass Function.
A: The Weierstrass function mentioned in Jesse Madnick's answer is the standard example, but I think this example is slightly misleading. The fact that it is constantly presented as the standard example may suggest that such examples are rare and must be constructed in a certain way. Actually such examples are extremely common; in an appropriate sense, the "generic" continuous function is nowhere differentiable. 
To my mind, the point of the Weierstrass function as an example is really to hammer in the following points:


*

*The uniform limit of continuous functions must be continuous, but

*The uniform limit of differentiable functions need not be differentiable.


However, if $f_n(x)$ is a uniformly convergent sequence of differentiable functions such that the derivatives $f_n'(x)$ also converge uniformly, then the uniform limit $f(x)$ is differentiable, and $f'(x)$ is the uniform limit of the functions $f_n'(x)$. So what fails in the example of the Weierstrass function is that the derivatives do not even come close to converging uniformly. 
