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.

Mon May 5 12:53:33 CDT 1997