Let f:[a,b]→R be a continuous function on a compact interval of the real line. Suppose that f is differentiable almost everywhere and that f′(x)≥0 at every point of differentiability. Is it true that f is non-decreasing on [a,b]?
(If f is 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′≥0 at every point of differentiability, but g is not non-decreasing.
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