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$\Omega=\{(x,y)\in \mathbb{R^2}: 0<|x|<1,0. $u:\Omega \rightarrow \mathbb{R}$ be defined by: \begin{align} u(x,y) = \begin{cases} 1, & \mbox{x > 0} \\ 0, & \mbox{x < 0} \end{cases} \end{align}

$u$ cannot be approximated by any squence of functions in $C^\infty(\overline{\mathbb{\Omega}})\cap W^{1,p}$.

My approach:

$u$ is in $L^1(\Omega)$ and $u$ is $C^{\infty}(\Omega)$, because the region on which $u$ is non differentiable$(x=0)$ is not in the domain. $\therefore$ $u$ is in $W^{1,p}$ and in $C^{\infty}(\Omega)$. From here how do I show the above?

P.S. Do I show that there exists no sequence of functions in $C^{\infty}(\overline{\Omega})$ that tends to $u$?

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    When you say "approximated by", what topology do you mean? The same in your last sentence when you say "tends to". – 2017-02-14
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    In the last reference ie the "tends to" one, I think it should be the $L^1$ norm. – 2017-02-14

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