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Assume that $f:(a,b) \to R$ is a convex function. We know that $f$ is continuous, $f'$ exists except at most in a countably infinite set and $f'$ is increasing.

What about the viceversa? If $f$ is continuous, $f'$ exists except at most in a countably infinite set and $f'$ is increasing, then $f$ is convex?

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    $\begingroup$ Consider $F(x) := f(a) + \int_a^x f'(t) \lambda^1(dt)$. Then $f = F$ and $F$ is convex. $\endgroup$
    – Dieter Kadelka
    Apr 11, 2019 at 8:42

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