  For a symplectic twist map, we prove that there is a choice of weak K.A.M.
solutions that depend in a continuous way on the cohomology class. We thus
obtain a continuous function $u(\theta, c)$ in two variables: the angle
$\theta$ and the cohomology class $c$. As a result, we prove that the
Aubry-Mather sets are contained in pseudographs that are vertically ordered by
their rotation numbers. Then we characterize the $C^0$ integrable twist maps in
terms of regularity of $u$ that allows to see $u$ as a generating function. We
also obtain some results for the Lipschitz integrable twist maps. With an
example, we show that our choice is not the so-called discounted one (see
\cite{DFIZ2}), that is sometimes discontinuous. We also provide examples of
`strange' continuous foliations that cannot be straightened by a symplectic
homeomorphism.

Dynamical Systems;Symplectic Geometry