For massive multiple-input multiple-output (MIMO) systems, downlink channel state information (CSI) compression and feedback are crucial for enhancing system performance. Deep learning (DL)-based methods have been designed and proven to perform well in this task. However, the distribution of CSI in real-world communication systems may differ from the one observed during model training, which can undermine the effectiveness of DL-based methods due to their limited generalization ability. Several methods have been proposed to facilitate online training and enable network adaptation to unknown scenarios. Nevertheless, the knowledge learned from previous scenarios is often forgotten, leading to performance degradation when encountering a previous scenario again. In this letter, we propose a novel continuous learning-based CSI feedback approach, which can effectively address the challenge of catastrophic forgetting and ensure consistent high performances across all historical scenarios, thereby enhancing the generalization capability of the model.