We extend the metric proof of the inverse Lyapunov Theorem, given in [13] for continuous multivalued dynamics, by means of tools issued from weak KAMtheory, to the case where the set-valued vector field is just upper semicontinuous. This generality is justified especially in view of application to discontinuous ordinary differential equations. The more relevant new point is that we introduce, to compensate the lack of continuity, a family of perturbed dynamics, obtained through internal approximation of the original one, and perform some stability analysis of it.