Density in $W^{1,p}(\mathbb{R}^N)$ and in $W^{1,p}(\Omega)$

I know that $C^\infty(\mathbb{R}^N) \cap W^{1,p}(\mathbb{R}^N)$ and $C^\infty_c (\mathbb{R}^N)$ are dense in $W^{1,p}(\mathbb{R}^N)$ for all $1\le p < \infty$.

I also know that $C^\infty(\Omega) \cap W^{1,p}(\Omega)$ is dense in $W^{1,p}(\Omega)$, for all $1\le p<\infty$ and with $\Omega \subset \mathbb{R}^N$ general open set.

I studied an example that show $C^\infty_c(\Omega)$ is not dense in $W^{1,\infty}(\Omega)$ give taking $\Omega=(-1,1)\subset \mathbb{R}^N$, and $u(x)=|x|$. I was searching an example to see that $C^\infty(\mathbb{R}^N) \cap W^{1,\infty}(\mathbb{R}^N)$ is not dense in $W^{1,\infty}(\mathbb{R}^N)$