A Lyapunov-like characterization of robustness of pointwise asymptotic stability for differential inclusions

Given a dynamical system, pointwise asymptotic stability (a.k.a. semistability) of a set requires that every point in the set be a Lyapunov stable equilibrium, and that every solution converge to one of the equilibria in the set. This note shows that robustness of this property, for a compact set in a setting of a differential inclusion subject to standard basic assumptions, can be equivalently characterized by the existence of a continuous set-valued Lyapunov function.