Process monitoring and control requires detection of structural changes in a data stream in real time. This article introduces an efficient sequential Monte Carlo algorithm designed for learning unknown changepoints in continuous time. The method is intuitively simple: new changepoints for the latest window of data are proposed by conditioning only on data observed since the most recent estimated changepoint, as these observations carry most of the information about the current state of the process. The proposed method shows improved performance over the current state of the art. Another advantage of the proposed algorithm is that it can be made adaptive, varying the number of particles according to the apparent local complexity of the target changepoint probability distribution. This saves valuable computing time when changes in the changepoint distribution are negligible, and enables re-balancing of the importance weights of existing particles when a significant change in the target distribution is encountered. The plain and adaptive versions of the method are illustrated using the canonical continuous time changepoint problem of inferring the intensity of an inhomogeneous Poisson process, although the method is generally applicable to any changepoint problem. Performance is demonstrated using both conjugate and non-conjugate Bayesian models for the intensity. Appendices to the article are available online, illustrating the method on other models and applications.