In the wikipedia article, it states that Minkowskis Question Mark function is continuous but not absolutely continuous.
let $[a_0,a_1....]$ denote the continued fraction representation of a number $x$
$?(x)=a_0+2\sum_{n=1}^{m}\frac{(-1)^{n+1}}{2^{a_1+...+a_n}}$ if $x$ is rational and $?(x)=a_0+2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2^{a_1+...+a_n}}$ if $x$ is irrational.
For continuity, could we just define $?_n(x)=1/n+a_0+2\sum_{n=1}^{m}\frac{(-1)^{n+1}}{2^{a_1+...+a_n}}$ and use Weirstrass M-test? The sequence of functions clearly converges to $?(x)$ as $n\rightarrow \infty$, and by using $M_n=5/2$ $\forall n$ we have that $?(x)$ is continuous.
For absolute continuity, I am thinking the best way to show $?(x)$ is not absolutely continuous is to use the Luzin N property, where if we have $E\subset [0,1]$, with $\mu(E)=0$, then $\mu(?(E))=0$ where $\mu$ is the Lebesgue measure. The issue I am having though is finding such an $E$. The most obvious choice is $E=\mathbb{Q}\cap [0,1]$, but $\mu(?(E))=0$, since rationals are mapped to dyadic rationals, which have measure zero because they are countable.
Some other approaches could be to somehow use the fact that $?(x)-x$ is periodic with period one, or consider $?(C)$ where $C$ is the ternary cantor set.