TITLE: $f(x)$ and $h(x)$ are absolutely continuous functions. Is $e^{f(x)} |h(x)|$ as well?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: $e^x$ is not Lipschitz on $\mathbb{R}$, but is Lipschitz on any bounded interval (with a Lipschitz constant that depends on the interval). Since $f$ is continuous on $[0,1]$ it is bounded (I assume that you know this); let's say that $|f(x)|\le M$ for all $x\in[0,1]$. The exponential function is Lipschitz continuous on $[-M,M]$, so that $e^{f(x)}$ is absolutely continuous on $[0,1]$.
Forget about the following line. I was thinking in uniform continuity, not absolute continuity.
An alternative way: every continuous function on a bounded closed interval is absolutely uniformly continuous.