Recent advances in wireless communications and positioning devices have generated a tremendous amount of interest in the continuous monitoring of spatial queries. However, such applications can incur a heavy burden on the data owner (DO), due to very frequent location updates. Database outsourcing is a viable solution, whereby the DO delegates its database functionality to a service provider (SP) that has the infrastructure and resources to handle the high workload. In this framework, authenticated query processing enables the clients to verify the correctness of the query results that are returned by the SP. In addition to correctness, the dynamic nature of the monitored data requires the provision for temporal completeness, i.e., the clients must be able to verify that there are no missing results in between data updates. This paper constitutes the first work that deals with the authentication of continuous spatial queries, focusing on ranges. We first introduce a baseline solution (BSL) that achieves correctness and temporal completeness, but incurs false transmissions; that is, the SP has to notify clients whenever there is a data update, even if it does not affect their results. Then, we propose CSA, a mechanism that minimizes the processing and transmission overhead through an elaborate indexing scheme and a virtual caching mechanism. Finally, we derive analytical models to optimize the performance of our methods, and evaluate their effectiveness through extensive experiments.