Objective: Epilepsy, a prevalent neurological disease, demands careful diagnosis and continuous care. Seizure detection remains challenging, as current clinical practice relies on expert analysis of electroencephalography, which is a time-consuming process and requires specialized knowledge. Addressing this challenge, this paper explores automated epileptic seizure detection using deep learning, focusing on personalized continual learning models that adapt to each patient's unique electroencephalography signal features, which evolve over time. Methods: In this context, our approach addresses the challenge of integrating new data into existing models without catastrophic forgetting, a common issue in static deep learning models. We propose EpiSMART, a continual learning framework for seizure detection that uses a size-constrained replay buffer and an informed sample selection strategy to incrementally adapt to patient-specific electroencephalography signals. By selectively retaining high-entropy and seizure-predicted samples, our method preserves critical past information while maintaining high performance with minimal memory and computational requirements. Results: Validation on the CHB-MIT dataset, shows that EpiSMART achieves a 21% improvement in the F1 score over a trained baseline without updates in all other patients. On average, EpiSMART requires only 6.46 minutes of labeled data and 6.28 updates per day, making it suitable for real-time deployment in wearable systems. Conclusion:EpiSMART enables robust and personalized seizure detection under realistic and resource-constrained conditions by effectively integrating new data into existing models without degrading past knowledge. Significance: This framework advances automated seizure detection by providing a continual learning approach that supports patient-specific adaptation and practical deployment in wearable healthcare systems.