Function example? Continuous everywhere, differentiable nowhere [duplicate]

Solution 1:

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.

Solution 2:

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...).