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.