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This paper considers the problem of safety verification for discretetime, possibly discontinuous dynamical systems. Typical solutions rely on finding invariant sets or Lyapunov functions and require solving optimization problems, which suffer from scalability and numerical solvers issues. Recently, a δ-sampling method for verifying invariance for Lipschitz continuous dynamics was proposed, which does not make use of optimization. In this work we present a δ-sampling verification theorem that extends the previous result to general discrete-time, possibly discontinuous dynamics. This opens up the application of δ-sampling verification to hybrid systems. Moreover, this paper proposes verification of stability on a set by jointly verifying (finite-step) Lyapunov type functions on an annulus with a (finite-step) Lyapunov function on the inner hole. We further indicate that δ-sampling can also be used to verify Lyapunov conditions on the annulus. Lastly, we employ finite-step invariant sets and finite-step Lyapunov functions, respectively, together with δ-sampling to achieve more practical safety verification methods.

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