I am struggling with Evans PDE problem 5.10. #12. Here is the question: Let $V\subset \subset U$ be open sets. Show by example that if we have $\|D^hu\|_{L^1(V)}\le C$ for all $0< |h|<\frac{1}{2}\operatorname{dist}(V,\partial U)$, it does not necessarily follow that $u\in W^{1,1}(V)$.
I have no idea why the bigger open set $U$ is relevant. I would very much appreciate your help.