  In this paper, we explain some facts on the discrete case of weak KAM theory.
In that setting, the Lagrangian is replaced by a cost $c:X\times X \to
\mathbb{R}$, on a "reasonable" space $X$. This covers for example the case of
periodic time-dependent Lagrangians. As is well known, it is possible in that
case to adapt most of weak KAM theory. A major difference is that critical
sub-solutions are not necessarily continuous. We will show how to define a
Ma\~ne potential. In contrast to the Lagrangian case, this potential is not
continuous. We will recover the Aubry set from the set of continuity points of
the Ma\~ne potential, and also from critical sub-solutions.
