In the Wikipedia article on Sobolev spaces, it mentions that if $\Omega\subset \mathbb{R}$ is an open set, it is possible that $f\in W^{1,p}(\Omega)$ but the extension $Ef = \begin{cases}f\text{ on }\Omega\\ 0\text{ otherwise}\end{cases}$ might not be in $W^{1,p}(\mathbb{R})$.
I am suspecting this is because we can have the existence of a function $g\in L^p$ such that $\int f\phi'=\int g \phi$ for all $\phi$ with compact support in $\Omega$, but we couldn't find such a $g\in L^p$ such that $\int f\phi'=\int g \phi$ for all $\phi$ with compact support in the the whole $\mathbb{R}$. But I am having a hard time finding concrete examples.