An example in $H^1$ but not continuous other than a singular function

Question

Is there any function in $H^1(\Omega)$ but not in $C(\Omega)$ other than singular functions? (Given the dimension $N\geq2$)

For an example, assume $N\geq 3$ and let $\Omega:=\{x\in\mathbb{R}^N;|x|<1\}$, then define a function almost everywhere in $\Omega$ as: $$ u(x):=|x|^{-\lambda},\quad x\neq0 $$ where $0<\lambda<(N-2)/2$. Then $u\in H^1(\Omega)$ and $u\notin C(\Omega)$.

However, could I find another example such that the function is also not singular at some points?