If $f$ is absolutely continuous and $g$ is Lipschitz, then $f \circ g$ is absolutely continuous.

Question

I am able to prove the case when $f$ is Lipschitz and $g$ is absolutely continuous. But I am having trouble solving the problem as stated. The main problem I am facing is that if $\{(a_i, b_i)\}_i$ is a finite set of disjoint open intervals, $\{(g(a_i), g(b_i)\}_i$ may not be a disjoint, which prevents me in using the absolute continuity of $f$.

Can anyone give me a hint in resolving this? Thank you in advance.

Answer 1

Hint:

Lipschitz (assume global) means for any $a, b$ $$ |g(a_i)-g(b_i)|<K|a_i-b_i|<\delta $$ So it doesn't matter whether $\{(g(a_i), g(b_i)\}_i$ is disjoint or not. If not disjoint, $2$ intervals merge into one anyway.