<p>This paper focuses on stability analysis of linear systems with a time-varying delay. The delay is assumed to periodically vary within a closed interval. By partitioning the whole interval into monotonically increasing and decreasing subintervals, a novel discontinuous Lyapunov functional is constructed within the framework of discontinuous functionals. This is achieved by introducing additional state vectors and discontinuous looped-functional terms. New stability criteria are consequently developed that could effectively reduce the inherent conservatism involved in traditional approaches. Finally, a specific numerical example is provided to demonstrate the efficiency and advantages of the proposed method.</p>