On the interval [0,1]. Define $f(x)=x\sin(1/x)$ for $x\in(0,1]$ and $f(0)=0$. I didn't work out the exact details but I'm pretty sure that then $$\Big |\int_0^xf'(t)dt\Big |=\infty,$$ due to a process similar to something of the form $1-2+3-4+5-...$ , as one approaches zero from above.
However according to the measure-theoretic definition of absolute continuity, there should in fact be some set of measure zero $E\in[0,1]$ such that $$\Big |\int_Ef'd\mu\Big | > 0.$$
I wasn't under the impression that this was even possible.
Edit: Maybe I wasn't clear about what my question is. What I want is a proof (constructive or not) that there exists a set of measure zero $E$ such that $\Big |\int_Ef'd\mu\Big | > 0.$ Or if that's not possible then for someone to explain to me what my misconception is concerning the measure theoretic definition of absolute continuity:
If $\mu(E)=0$ then $v(E)=0$.
definition can also be found in Royden's Real Analysis