This paper focuses on semistability and finite-time semistability for discontinuous dynamical systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we extend the theory of semistability to discontinuous autonomous dynamical systems. In particular, Lyapunov-based tests for sernistability and finite-time semistability for autonomous differential inclusions are established.