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?
Best Answer
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.