TITLE: Composition of two absolute functions
QUESTION [7 upvotes]: $f$ and $g$ are two absolutely contiunous functions and $f$ is monotone.
Is $f(g)$ necessarily an absolutely continuous function? Why not any counter example?
Thanks a lot!!

REPLY [14 votes]: Same idea as @AlbertH, but a bit simpler to write: $f(x)=x^2 \sin^2 \frac{\pi }{2x}$ is Lipschitz on $[0,1]$ because $f'$ is bounded. Hence, $f$ is absolutely continuous. However, $\sqrt{f}$ has infinite variation on $[0,1]$, which one can demonstrate by summing $|f(\frac1n)-f(\frac1{n+1})|$.