Is composition of an absolutely continuous function and a $C^1$ function still absolutely continuous?

Question

Suppose $y:[a,b]\to \mathbb{R}^n$ is absolutely continuous, where $[a,b]\subset \mathbb{R}$ is a compact interval. Let $\phi:\mathbb{R}^n\to \mathbb{R}$ be a $C^1$ function. Is it true that $$\phi \circ y:[a,b]\to \mathbb{R}$$ is also absolutely continuous?

Can I have an hint in order to prove/disprove it?

Thanks a lot in advance.

Answer 1

Hint: since $y([a,b])$ is compact and connected, the restriction of $\varphi$ to $y([a,b])$ is Lipschitz. What can you say about the composition $\psi \circ y$ if $\psi$ is Lipschitz?