Let's suppose $N\geq 2$. I know that functions in $W^{1,p}(\mathbb{R}^N)$ (with $p\leq N$) can be discontinous but all the examples that I know are not jump discontinuities (I've explained in the comments what I mean with jump discontinuities). Are there any examples of such discontinuities? And what about the case of a regular domain $\Omega\subseteq \mathbb{R}^N$ (a bounded open set with $C^1$ boundary)?
I've hard that Sobolev functions "don't like to jump", but I don't know any rigorous statement about this. Could you help me?