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In this paper, a feedback stabilization approach is proposed for a class of time-delay systems with discontinuity based on functional differential inclusions. By extending the differential inclusion in the sense of Filippov to functional differential inclusion, a notion of the solution for time-delay systems with discontinuity is presented that is defined satisfying the proposed Filippov set-valued functional. With this notion, it is shown that both the Lyapunov-Krasovskii stability theorem and the LaSalle invariance principle can be easily extended to discuss the stability problem of time-delay systems with discontinuity. Based on the introduced analysis tools, it is proved that a feedback stabilization controller can be obtained for a class of nonlinear time-delay systems with discontinuity such that the closed loop system is strongly asymptotically stable. Simulation result of a numerical example is given to demonstrate the proposed control approach.