Let $\Omega=\{(x, y), 0<|x|<1,0<y<1\} \subset \mathbb{R}^{2}$. Define the function $$ u(x, y)= \begin{cases}1 & \text { if } x>0 \\ 0 & \text { if } x<0\end{cases} $$ (a) Show that $u \in W^{1, p}(\Omega), \forall p \geqslant 1$.
(b) Show that there exists a $\varepsilon>0$, such that there is no function $\phi \in C^{1}(\bar{\Omega})$ such that $\|u-\phi \|_{1, p}<\varepsilon$. What interpretation does this result allow?
I was able to do item (a), my question is regarding item (b). Any ideas or hints?