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Let $f : [0,1] \to \mathbb{R}$ be a Holder-continuous function of an exponent $\alpha \in (0,1)$ and differentiable a.e. at the same time.

Assume further that the derivative $f'$ is integrable on $[0,1]$.

Then I wonder if $f$ is absolutely continuous as well.

Holder-continuity excludes counterexamples like Cantor functions, I believe. But I cannot proceed further.

Could anyone help me?

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The Cantor function as defined in its Wikipedia page is Hölder-continuous for $\alpha = \frac{\ln 2}{\ln 3}$ (a proof can be found here for example: Proof Clarification for Cantor Function is Holder Continuous), almost everywhere differentiable, of derivative $0$ almost everywhere thus of integrable derivative, but is not absolutely continuous.

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  • $\begingroup$ Oh my.....it is so surprising and against intuition....Thank you so much for your answer.... $\endgroup$
    – Keith
    Jul 24, 2023 at 20:49

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