Given a function $f(x)=\int_{0}^{g(x)}h'(t)dt$, where $h$ is absolutely continuous and $g$ is differentiable everywhere, can we say that $f(x)$ is absolutely continuous? If not, what further conditions on $g$ and $h$ are needed?
The absolute continuity of $h$ here is allowing us to write $h(c)=\int_{0}^{c}h'(t)dt$ for fixed $c$. The question is how much smoothness must $g(x)$ have with respect to $x$ to make $f(x)$ absolutely continuous, if we now allow the upper boundary $c$ of the integral to move according to $g(x)$. We know that $f(x)$ is differentiable almost everywhere iff it is absolutely continuous, so Liebniz rule for differentiating the integral suggests that $h'(g(x))g'(x)$ existing almost everywhere is sufficient. This is guaranteed by the absolute continuity of $h$ and $g$ being differentiable. But I am not sure the application of Liebniz rule here is allowable.