Weierstrass' nowhere differentiable function.
Below are parts of the graph of the function
Weierstrass showed that this function is continuous everywhere, but
that you can't differentiate it at any x.
His proof is quite complicated, but the following pictures give you an
impression of what the function does.
There are three pictures.
The first is of the
graph at a normal scale (x ranges from 0 to about 4).
In the second you see a piece of the graph
magnified by about 100.
In the third the graph is
magnified 10,000 times.