  For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in
\mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost
everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise
affine away from a closed, countably $\mathcal{H}^{n - 1}$-rectifiable set. If
$A$ is affinely independent, then $u$ is piecewise affine away from a closed
$\mathcal{H}^{n - 1}$-null set.
