Almost everywhere differentiable function with continuous derivative
When you are learning real analysis it is wise to remember a few famous counterexamples. I think that it is safe to say that the Cantor function is probably the most famous.
It gives you a surprising example of a nonconstant continuous function with a zero derivative almost everywhere. It destroys your question and has destroyed many a bad conjecture in the past.
https://en.wikipedia.org/wiki/Cantor_function