Gossip-based Peer-to-Peer(P2P) streaming has been proved to be an effective and resilient method to stream qualified media contents in dynamic and heterogeneous network environments. Because of the intrinsic randomness of gossiping, some data segments cannot be disseminated to every node in time, which seriously affects the media playback continuity. In this paper we describe the design of ContinuStreaming, a gossip-based P2P streaming system which can maintain high resilience and low overhead while bring a novel and critical property — full coverage of the data dissemination. With the help of DHT, data segments which are likely to be missed by the gossip-based data scheduling can be quickly fetched by the on-demand data retrieval so as to guarantee continuous playback. We discuss the results of both theoretical analysis and comprehensive simulations on various real-trace overlay topologies to demonstrate the effectiveness of our system. Simulation results show that ContinuStreamin...