The holy grail of machine learning is to enable Continual Federated Learning (CFL) to enhance the efficiency, privacy, and scalability of AI systems while learning from streaming data. The primary challenge of a CFL system is to overcome global catastrophic forgetting, wherein the accuracy of the global model trained on new tasks declines on the old tasks. In this work, we propose Continual Federated Learning with Aggregated Gradients (C-FLAG), a novel replay-memory based federated strategy consisting of edge-based gradient updates on memory and aggregated gradients on the current data. We provide convergence analysis of the C-FLAG approach which addresses forgetting and bias while converging at a rate of $O(1/\sqrt{T})$ over $T$ communication rounds. We formulate an optimization sub-problem that minimizes catastrophic forgetting, translating CFL into an iterative algorithm with adaptive learning rates that ensure seamless learning across tasks. We empirically show that C-FLAG outperforms several state-of-the-art baselines on both task and class-incremental settings with respect to metrics such as accuracy and forgetting.