Find a weak differentiable function which is discontinuous

Question

I'm seeking a weak differentiable function $f \in W^{1,1}(\Omega)$ which is discontinuous. I think every weak differentiable function defined on $\mathbb{R}$ is continuous by Sobolev embedding theorem, so an example should exist at least in $\mathbb{R}^2$ if it does. Is my guess correct? If so, could anyone please show me an example?

Answer 1

You require an integrable function that is absolutely continuous on almost all lines parallel to the coordinate axes, whose partial derivatives are also integrable. Try $f(x) = \dfrac 1{\sqrt{|x|}}$ and $\Omega = B(0,1)$ in $\mathbb R^2$.

Answer 2

Yes, you are right. On $\mathbb{R}$ you will have no such luck. An example in higher dimensions is relatively easy: Take $f(x) = |x|^s$ where $s > 1- n$. In fact you can take any $C^1$ function in higher dimensions (on bounded domain for integrability) and just make it discontinuous in one point. This function will still have a weak derivative that is bounded.