I have shown that all Lipschitz functions are absolutely continuous, which is easy to show, but I am in doubt as to whether any function that meets Holder's condition is absolutely continuous, I don't know if I am restricting myself to a certain $\alpha$ in particular.
attempt Let $f\in \mathcal{C^{\alpha}}([a,b])$, and let a collection $\lbrace (a_{k},b_{k})\rbrace_{k=1}^{n}$ disjoint intervals of $[a,b]$, since $\varepsilon$, there exist, such that:
$\displaystyle \sum_{k=1}^{n}|f(b_{k})-f(a_{k})|\leq \sum_{k=1}^{n} K|b_{k}-a_{k}|^{\alpha}<K\left[\left(\frac{\varepsilon}{K}\right)^{\frac{1}{\alpha}} \right]^{\alpha}=\varepsilon$
Hence $f$ is absolutely continuous, if $\displaystyle \sum_{k=1}^{n}(b_{k}-a_{k})<\left(\frac{\varepsilon}{K}\right)^{\frac{1}{\alpha}}$.
It is correct to affirm for any $\alpha>0$