  We study the two dimensional least gradient problem in convex polygonal sets
in the plane, $\Omega$. We show the existence of solutions when the boundary
data $f$ are attained in the trace sense. The main difficulty here is a
possible discontinuity of $f$. Moreover, due to the lack of strict convexity of
$\Omega$, the classical results are not applicable. We state the admissibility
conditions on the boundary datum $f$, that are sufficient for establishing an
existence result. One of them is that $f\in BV(\partial\Omega)$. The solutions
are constructed by a limiting process, which uses solutions to known problems
