Then $u = u_c = \arg(z) \in BV(\Omega) \cap C^{\infty}(\Omega)$, but $u$ isn’t continuous at $0 \in \partial\Omega$.

Existence of solutions on the plane
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We shall prove existence of solutions on the plane for boundary data in $BV(\partial\Omega)$. We are going to use approximations of the solution in strict topology. Proposition \[podciagzbiezny\] will ensure us that existence of convergent sequences of approximations in $L^1$ topology is not a problem; Theorem \[tw:scislazb\] will upgrade it to strict convergence. The Miranda stability theorem (Theorem \[stabilnosc\]) ends the proof. Later, we shall see an example of a discontinuous function $f$ of infinite total variation such that the solution to the LGP exists.

\[podciagzbiezny\] Suppose $f_n \rightarrow f$ in $L^1(\partial\Omega)$. $u_n$ are solutions of LGP for $f_n$. Then $u_n$ has a convergent subsequence, i.e. $u_{n_k} \rightarrow u$ in $L^1(\Omega)$.

As the trace operator is a surjection, by the Open Mapping Theorem it is open. Let us fix $\widetilde{f} \in BV(\Omega)$ such that $T\widetilde{f} = f$ and a sequence of positive numbers $\varepsilon_n \rightarrow 0$. Then by continuity and openness of $T$ the image of a ball $B(\widetilde{f}, \varepsilon_n)$ contains a smaller ball $B(T\widetilde{f}, \delta_n)$ for another sequence of positive numbers $\delta_n \rightarrow 0$. 