The problem: Let $q_n$ be an enumeration of $B_1(0)\cap \mathbb Q^2$. Show that the function $$B_1(0)\to \mathbb R : x\mapsto \sum_n 2^{-n} \frac{1}{\sqrt{\|x-q_n\|}}$$ is in the Sobolev Space $W^{1,1}(B_1(0))$ and that every representative of it's class in $L^1$ is nowhere continuous.
My approach: Actually, I am getting a bit confused in showing that this function has a first order weak derivative and that is again integrable. I can't proceed to find the derivative of this map.
And there's a hint given to show the second part:
Show that for every open set $U\subset B_1(0)$, the essential supremum of the function restricted to $U$ is unbounded.
I need hints for the first question. Then maybe I can get some leads to prove the second statement.
Thanks in advance for any hints.