$L^1_{loc}(\mathbb{R}^N)$ as $r \rightarrow 0$.

It turns out that on the plane Theorem \[twierdzeniezbgg\] may be improved to an analogue of the maximum principle for linear equations; geometrically speaking, the linear weak maximum principle states that every level set touches the boundary.

\[slabazasadamaksimum\] $($weak maximum principle on the plane$)$\
Let $\Omega \subset \mathbb{R}^2$ be an open bounded set with Lipschitz boundary and suppose $u \in BV(\Omega)$ is a function of least gradient. Then for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is empty or it is a sum of intervals, pairwise disjoint in $\Omega$, such that every interval connects two points of $\partial \Omega$.

By the argument from [@Giu Chapter 10] for every $t \in \mathbb{R}$ the set $\partial \{ u > t \}$ is a sum of intervals and $\partial \{ u > t \} = \partial^* \{ u > t \}$. Obviously $\partial \{ u > t \}$ is closed in $\Omega$. Suppose one of those intervals ends in $x \in \Omega$. Then the normal vector at $x$ is not well defined (the statement of the Theorem \[tw:blowup\] does not hold), so $x \notin \partial^* \{ u > t \}$. Thus $x \notin 