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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 thusobtain a continuous function $u(\theta, c)$ in two variables: the angle$\theta$ and the cohomology class $c$. As a result, we prove that theAubry-Mather sets are contained in pseudographs that are vertically ordered bytheir rotation numbers. Then we characterize the $C^0$ integrable twist maps interms of regularity of $u$ that allows to see $u$ as a generating function. Wealso obtain some results for the Lipschitz integrable twist maps. With anexample, 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 symplectichomeomorphism.

On the transversal dependence of weak K.A.M. solutions for symplectic twist maps