So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. Let $S_u,S_{u_k}$ be the approximate discontinuity sets of $u,u_k$ respectively and fix $x_0\in \Omega_1:=\Omega \setminus \left(S_u\cup S_{u_1}\cup\cdots\right)$. The approximate limits $\tilde u$ and $\tilde u_k$ are well-defined in $\Omega_1$ (I am following the notation of Ambrosio-Fusco-Pallara). Does it hold that $\tilde u_k(x_0)\rightarrow \tilde u(x_0)$, possibly up to a subsequence?

I think that Egorov's theorem has to come up at some point but I am not able to include $x_0$ to the set where the functions converge uniformly.

Moreover, I am not sure if it is of any help, but we may assume that $u,u_k$ are pointwise uniformly bounded (ex. they have values in $[0,1]$).

Any help is warmly welcome!