Non-continous Sobolev function in $W^{1,2}(B_1)$ for $B_1$ the unit ball in $\mathbb{R}^2$

Question

I am currently studying for my Functional Analysis 2 exam and for a better understanding of Sobolev spaces I wanted to write down a function $u \in W^{1,2}(B_1)$ (for $B_1 \subseteq \mathbb{R}^2$ the unit ball in $\mathbb{R}^2$) which is not continous. Unfortunately I could not come up with any such function.

Thanks for your help!

Answer 1

You coud try the function $f(x)=\log(-\log|x|)$.