In this paper, we investigate how consistency can be established for replicated applications changing their state in reaction to user-initiated operations as well as the passing of time. Typical examples of these applications are networked computer games and distributed virtual environments. We give a formal definition of the terms consistency and correctness for this application class. Based on these definitions, it is shown that an important tradeoff relationship exists between the responsiveness of the application and the appearance of short-term inconsistencies. We propose to exploit the knowledge of this tradeoff by voluntarily decreasing the responsiveness of the application in order to eliminate short-term inconsistencies. This concept is called local-lag. Furthermore, a timewarp scheme is presented that complements local-lag by guaranteeing consistency and correctness for replicated continuous applications. The computational complexity of the timewarp algorithm is determined in theory and practice by examining a simple networked computer game. The timewarp scheme is then compared to the well-known dead-reckoning approach. It is shown that the choice between both schemes is application-dependent.