Are there $H^1_0(\Omega)$-functions in the plane that are discontinuous over curves?

Question

Consider a bounded domain $\Omega \subset \mathbb{R}^2$ with Lipschitz-boundary and a curve $\gamma : [0,1] \rightarrow \Omega$. Is it possible to construct a function $f \in H^1_0(\Omega)$ which is discontinuous over $\gamma([0,1])$ ($\lim_{y \ \searrow \ x}{f(x)} \neq \lim_{y \ \nearrow \ x}{f(x)} \ \forall x \in \gamma([0,1])$)?

Answer 1

The idea is to construct $f$ as a linear combination of translations of the function $g$ given by $$ g(x) = \log |\log (|x|)|\phi(x), $$ where $\phi\in C_0(B_{1/2}(0))$ is a smooth cut-off function with $\phi(x)\ge1$ for $|x|<\epsilon$, where $\epsilon\in (0,1/2)$ is given.

Now take the curve $\gamma$, take a countable, dense subset $(\tau_k)\in(0,1)$. Then set $$ f(x) := \sum_{k=1}^\infty 2^{-k} g(x-\gamma(\tau_k)). $$ Then $f$ is unbounded in every open neighborhood of points on the curve. Hence the left/right-sided limits do not even exist.