Functions in $H^1(\mathbb{R}^2)$ but not in $C(\mathbb{R}^2)$? [closed]

Question

I'm struggling to come up with examples of functions which are in $H^1(\mathbb{R}^2)$ but not in $C(\mathbb{R}^2)$. I know that $H^1(\mathbb{R})\subset C(\mathbb{R})$, and I know the power law example of a function in $H^1(\mathbb{R}^3)$ that's not in $C(\mathbb{R}^3)$, but the $\mathbb{R}^2$ case eludes me.

Answer 1

It is sufficient to focus on the unit ball $B$ in $\mathbb{R}^N$, $N\geq 2$. Consider the function $$ u(x)=\sqrt{1-|x|}\cdot\left( \frac{1}{|\ln(1-|x|)|}\right)^\alpha $$ with $1/2<\alpha<N/2$, which is singular at the origin.

PS. Note that the function is actually in $H^1_0(B)$.