This paper introduces a discontinuous rendezvous algorithm for multi-agent systems which allows a group of agents to achieve finite-time rendezvous with connectivity preservation as well as disturbance rejection. The proposed discontinuous rendezvous algorithm is based on a class of general communication functions with certain available information range. Nonsmooth stability analysis and graph theory are employed to obtain the finite-time rendezvous. An upper bound on the convergence time is also given. Our result simultaneously satisfies the three features: connectivity preservation, disturbance rejection and finite-time convergence. (C) 2016 Elsevier Ltd. All rights reserved.