Let $f \colon [a,b] \rightarrow \mathbb{R}$ be a continuous function on a compact interval of the real line. Suppose that $f$ is differentiable almost everywhere and that $f'(x) \geq 0$ at every point of differentiability. Is it true that $f$ is non-decreasing on [a,b]?
(If $f$ is $\textit{absolutely}$ continuous, this is certainly true, but I'm not so sure what happens if you weaken the assumption to mere continuity.)
Answer
No. The Cantor-Lebesgue function $f$ is continuous, non-decreasing, non-constant, and satisfies $f'=0$ almost everywhere. It's not hard to see that $f$ is non-differentiable at every point of the Cantor set. So $g=-f$ is a counterexample to your question: $g$ is continuous, differentiable almost everywhere, satisfies $g'\ge 0$ at every point of differentiability, but $g$ is not non-decreasing.
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