Question on monotonicity and differentiability Let $f:[0,1]\rightarrow \Re$ be continuous. Assume $f$ is differentiable almost everywhere and $f(0)>f(1)$.
Does this imply that there exists an $x\in(0,1)$ such that $f$ is differentiable at $x$ and $f'(x)<0$?
My gut feeling is yes but I do not see a way to prove it. Any thoughts (proof/counterexample)?
Thanks!

As stated in another answer the Cantor function is a counterexample.
You would need to assume differentiability for all $x \in (0,1)$. "Almost everywhere" is not good enough.
