Thursday, 1 March 2018

real analysis - Continuous function with non-negative derivative a.e. implies non-decreasing?



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 g0 at every point of differentiability, but g is not non-decreasing.


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