A friend of mine is covering the Cauchy-Riemann equations in his complex variables class and wondered if there is a real variable function that is differentiable at precisely one point.
The answer is “yes”, of course, but the example I could whip up on the spot is rather pathological.
Here is one example:
Let be defined as follows:
That is, if is irrational or zero, and is if is rational and where .
Let be given and choose a positive integer so that . Let . Now if and is irrational, then .
Now the fun starts: if is rational, then and .
We looked at the right hand limit; the left hand limit works in the same manner.
Hence the derivative of exists at and is equal to zero. But zero is the only place where this function is even continuous because for any open interval , .