# Sobolev differentiability of composite function

I was wondering about the following fact: if $\Omega$ is a bounded subset of $\mathbb{R}^n$ and $u\in W^{1,p}(\Omega)$ and $g\in C^1(\mathbb{R},\mathbb{R})$ such that $|g'(t)t|+|g(t)|\leq M$, is it true that $g\circ u \in W^{1,p}(\Omega)$?

If $g'\in L^{\infty}$ this would be true, but here we don't have this kind of estimate...

-

You get $g' \in L^\infty$ from the assumptions, since $|g'(t)| \le M/|t|$, and $g'$ is continuous at $0$.