Let $\Omega = B(0,1)$ be the open unit disc in $\mathbb{R}^2$. I'm looking for an example of a discontinuous and bounded function in $W^{1,2}(\Omega)$.
I know the example $u(x) = \log \left( \log \left(1 + \frac{1}{|x|}\right)\right)$ of a discontinuous but unbounded function in $W^{1,2}(\Omega)$. I've tried playing with things like $(x,y) \mapsto \frac{x}{(x^2 + y^2)^{1/2}}$ but it didn't get me far. Any insight on how to try and construct such examples and how to expect such functions to behave would be much welcomed!
One can get an example just by composing the function $u(x,y)$ with the function $f(x) = \sin(x)$. By some variant of a chain rule for Sobolev functions, a composition of function in $u \in W^{1,p}(\Omega)$ with a function $f \in C^1_B(\mathbb{R})$ results in a function in $W^{1,p}(\Omega)$. Choosing for $f(x)$ a bounded function that doesn't have a limit when $x \rightarrow \infty$ and composing it with an unbounded $u$ gives the required example.
Of course, the belonging of $f \circ u$ to $W^{1,2}(\Omega)$ can be easily checked directly.