In this study, we explore an emerging research area of Continual Learning for Temporal Sensitive Question Answering (CLTSQA). Previous research has primarily focused on Temporal Sensitive Question Answering (TSQA), often overlooking the unpredictable nature of future events. In real-world applications, it's crucial for models to continually acquire knowledge over time, rather than relying on a static, complete dataset. Our paper investigates strategies that enable models to adapt to the ever-evolving information landscape, thereby addressing the challenges inherent in CLTSQA. To support our research, we first create a novel dataset, divided into five subsets, designed specifically for various stages of continual learning. We then propose a training framework for CLTSQA that integrates temporal memory replay and temporal contrastive learning. Our experimental results highlight two significant insights: First, the CLTSQA task introduces unique challenges for existing models. Second, our proposed framework effectively navigates these challenges, resulting in improved performance.

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