if $f : [0,1] \to \mathbb{R}$ is differentiable almost everywhere and right-continuous almost everywhere, is it absolutely continuous?

Question

Let $f : [0,1] \to \mathbb{R}$ be a function defined everywhere and differentiable almost everywhere on $[0,1]$. Moreover, the derivative $f'$ is integrable.

Assume further that $f$ is right-continuous almost everywhere. That is, there exists $E \subset [0,1)$ of measure zero such that if $t \in [0,1)-E$, then \begin{equation} f(t+\epsilon) \to f(t) \text{ as } \epsilon \to 0^+ \end{equation}

Then, I wonder if these conditions are sufficient to conclude that $f$ is absolutely continuous on $[0,1]$.

Could anyone please help me?