I want to prove that every absolutely continuous function on $[a,b]$ admits weak derivative, and it coincides with its derivative a.e. .
So I pick $f\in AC([a,b])$, and $\phi\in \mathcal{C}^{\infty}_c((a,b))$: I want to prove $$\int_a^b f\phi'dx=-\int_a^b g\phi dx$$ where $g\in L^2((a,b))$.
My idea was to use integration by parts, from which it follows \begin{equation} \int_a^b f\phi'dx= (f\phi)\mid_a^b - \int_a^bf'\phi dx \end{equation}
We know $\phi$ vanishes at the boundary, thus we have (thanks to the fundamental lemma of the calculus of variations) that $g=f'$ a.e. and we are done.
My question: I proved that integration by parts holds if $f,\phi\in AC([a,b])$, and $\phi\in \mathcal{C}^{\infty}_c((a,b))$: is it true that any infinitely differentiable function is also absolutely continuous? Because I was thinking at $\phi(x)=e^x$, which is infinitely differentiable but not (I think) absolutely continuous (but in this case $e^x$ does not vanish at the boundary). I know uniformly continuous is not enough, as the Cantor-Lebesgue-Vitali function is uniformly continuous but not absolutely continuous.
Moreover since this question was inside my mind I'm having problems about the interval I'm considering (sometimes is open, sometimes is closed) and this clearly create confusion.
Any hint, help or reference would bu much appreciate, thanks in advance.