A function which is differentiable at a single point, and discontinuous everywhere else.

This is a slight variation on the previous example. Define f(x) by

As in the previous example, f is discontinuous everywhere except at x=0. This means that f certainly can't be differentiable for any , since differentiability implies continuity. But at x=0 it is not only continuous but also differentiable! Look at

The difference quotient can be written as

But we saw that this limit is 0 in the previous example (with x instead of h, but hey, it's just a letter). So, f'(0) exists and is 0.