# Composition of sobolev functions

The following materials are exercise 8.10 and 8.11 from the functional analysis book by Prof. Brezis. The questions arise from whether we can well define a composition of two $W^{1,p}$ functions or not. From Corollary 8.11 we know if $G \in C^1, u \in W^{1,p}$, $G(u) \in W^{1,p}$ is well defined. However bad things might happen if $G \in W^{1,p}$. For instance, $G(x) = |x|$, then $G(u)$ might be bad on the set $\{ u = 0\}$. I thought we could still define things in the sense of distribution, but Prof. Brezis said in this special case we could have something more than that, because of the following propositions. The proof is not hard by following the hints, but the idea is really clever (IMO)!