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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 tex2html_wrap_inline205 , 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.

Tom Vogel
Mon May 5 12:53:33 CDT 1997