[Math] Absolutely Continuous but not Holder continuous

Question

Can some one give me an example of a function which is Absolutely continuous but not Holder continuous?

Thanks

Answer 1

$$f(x)=\begin{cases} 1/\log x \quad &\text{if } x\in (0,1/2] \\ 0 &\text{if }x=0\end{cases}$$ Consider the behavior of $f(x)/x^\alpha$ at zero.

Also notice that $f'$ is bounded on $[1/n,1/2]$ for all $n=3,4,\dots$.