In wireless networks with integrated sensing and communications (ISAC), edge intelligence (EI) is expected to be developed at edge devices (ED) for sensing user activities based on channel state information (CSI). However, due to the CSI being highly specific to users' characteristics, the CSI-activity relationship is notoriously domain dependent, essentially demanding EI to learn sufficient datasets from various domains in order to gain cross-domain sensing capability. This poses a crucial challenge owing to the EDs' limited resources, for which storing datasets across all domains will be a significant burden. In this paper, we propose the EdgeCL framework, enabling the EI to continually learn-then-discard each incoming dataset, while remaining resilient to catastrophic forgetting. We design a transformer-based discriminator for handling sequences of noisy and nonequispaced CSI samples. Besides, we propose a distilled core-set based knowledge retention method with robustness-enhanced optimization to train the discriminator, preserving its performance for previous domains while preventing future forgetting. Experimental evaluations show that EdgeCL achieves 89% of performance compared to cumulative training while consuming only 3% of its memory, mitigating forgetting by 79%.