This work investigated predefined-time stability for discontinuous nonautonomous systems and its application to cluster synchronization in community networks with discontinuous intra-community interactions. First, a new predefined-time stability criterion for nonautonomous systems was established via differential inclusions and a generalized Lyapunov method, allowing indefinite time derivatives almost everywhere. Second, a community network model was introduced incorporating discontinuous intra-community interactions, time-varying parameters, and sign-dependent coupling to represent cooperative and competitive relationships. For such networks, a state-feedback control protocol was designed that, using the proposed stability result, ensures cluster synchronization within any prescribed time. A numerical example with sign-based coupling validated the theoretical analysis.