along $S_k$; in particular the jump of $u$ is constant along $S_k$.

We follow the characterisation of $J_u$ from the Proposition \[stw:zbiorskokow\]. For every $t$ the set $\partial^{*} \{ u > t \}$ is a minimal surface. Proposition \[stw:sternbergpowmin\] ensures that if $\partial^{*} \{ u > s \} \cap \partial^{*} \{ u > t \} \neq \emptyset$, then their intersecting connected components $S_{s,i}$, $S_{t,j}$ coincide. In particular, the trace from both sides defined as above is constant along $S_{t,j}$. Thus connected components of $J_u$ coincide with connected components of $\partial^{*} \{ u > t \}$ for some $t$, so by weak maximum principle they are minimal surfaces non-intersecting in $\Omega$ with boundary in $\partial \Omega$. As the area of each such surface is positive, there is at most countably many of them.

\[tw:rozklad\] Let $\Omega \subset \mathbb{R}^N$, where $N \leq 7$, be an open, bounded, strictly convex set with Lipschitz boundary. Suppose $u \in BV(\Omega)$ is a function of least gradient. Then there exist functions $u_c, u_j \in BV(\Omega)$ such that $u = u_c + u_j$ and $(Du)_c = Du_c$ and $(Du)_j = Du_j$, i.e. one can represent $u$ as a sum of a continuous function and a piecewise constant 