Given that functions $f(x)$ and $h(x)$ are absolutely continuous on $[0,1]$, I want to show that $e^{f(x)} |h(x)|$ is absolutely continuous as well.
I know that (1) the product of two absolutely continuous function on $[0,1]$ is absolutely continuous. (2) the composition of a Lipschitz continuous function and an absolutely continuous function is absolutely continuous. So $|h|$ is absolutely continuous,
But the exponential function $e^x$ is not Lipschitz, so not absolutely continuous.
What's the key to solve the problem here?