Given a continuous function $f\colon(0,1)\to\mathbb{R}$ which is differentiable almost everywhere in $(0,1)$ with $f'(x)=g(x)$ for almost all $x\in(0,1)$. Now suppose that $g$ is itself continuous. Can you give me a simple rationale why $f$ must be differentiable everywhere in $(0,1)$?
Almost everywhere differentiable function with continuous derivative
real-analysis
Best Answer
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