A function with a derivative defined for all x, but whose derivative is discontinuous.

This gets a little involved. Define f by

For we can find f'(x) by using the standard rules: it's . This doesn't have a limit as x goes to 0: if you look at the sequence (which goes to 0), , but looking at the sequence , . Therefore does not exist.

However, f'(0) does exist. Use the definition of derivative:

Since , the difference quotient satisfies

for all . But , so

by the pinching theorem. Therefore f'(0) exists and equals 0.

Here's a picture of f: