Continual learning approaches help deep neural network models adapt and learn incrementally by trying to solve catastrophic forgetting. However, whether these existing approaches, applied traditionally to image-based tasks, work with the same efficacy to the sequential time series data generated by mobile or embedded sensing systems remains an unanswered question. To address this void, we conduct the first comprehensive empirical study that quantifies the performance of three predominant continual learning schemes (i.e., regularization, replay, and replay with examples) on six datasets from three mobile and embedded sensing applications in a range of scenarios having different learning complexities. More specifically, we implement an end-to-end continual learning framework on edge devices. Then we investigate the generalizability, trade-offs between performance, storage, computational costs, and memory footprint of different continual learning methods. Our findings suggest that replay with exemplars-based schemes such as iCaRL has the best performance trade-offs, even in complex scenarios, at the expense of some storage space (few MBs) for training examples (1% to 5%). We also demonstrate for the first time that it is feasible and practical to run continual learning on-device with a limited memory budget. In particular, the latency on two types of mobile and embedded devices suggests that both incremental learning time (few seconds - 4 minutes) and training time (1 - 75 minutes) across datasets are acceptable, as training could happen on the device when the embedded device is charging thereby ensuring complete data privacy. Finally, we present some guidelines for practitioners who want to apply a continual learning paradigm for mobile sensing tasks.