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This paper deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential/difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Moreover, it is the starting point for solving the stabilization problem, namely for finding a state feedback controller which finite-time stabilizes the closed loop system. Some examples illustrate the effectiveness of the proposed approach.