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Result diversification has recently attracted considerable attention as a means of increasing user satisfaction in recommender systems, as well as in web and database search. In this paper, we focus on the problem of selecting the k-most diverse items from a result set. Whereas previous research has mainly considered the static version of the problem, in this paper, we exploit the dynamic case in which the result set changes over time, as for example, in the case of notification services. We define the CONTINUOUS k-DIVERSITY PROBLEM along with appropriate constraints that enforce continuity requirements on the diversified results. Our proposed approach is based on cover trees and supports dynamic item insertion and deletion. The diversification problem is in general NP-hard; we provide theoretical bounds that characterize the quality of our cover tree solution with respect to the optimal one. Since results are often associated with a relevance score, we extend our approach to account for relevance. Finally, we report experimental results concerning the efficiency and effectiveness of our approach on a variety of real and synthetic datasets.