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This paper deals with the stability of systems with discontinuous righthand side (with solutions in Filippovs sense) via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of "set-valued derivative" of vector Lyapunov functions is introduced, some generalized comparison principles on dis(c)ontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.