Epilepsy, a major neurological disease, requires careful diagnosis and treatment. However, the detection of epileptic seizures remains a significant challenge. Current clinical practice relies on expert analysis of EEG signals, a process that is time-consuming and requires specialized knowledge. This paper explores the potential for automated epileptic seizure detection using deep learning techniques, with a particular focus on personalized models based on continual learning. We highlight the importance of adapting these models to each patient's unique EEG signal features, which evolve over time. Our approach addresses the fundamental challenge of integrating new data into existing models without losing previously acquired information, a common issue in static deep learning models when applied in dynamic environments. In this study, we propose a novel continual learning algorithm for seizure detection, which integrates a replay buffer mechanism. This mechanism is key to retaining relevant information on past data while acquiring new one, thus effectively enhancing the model's performance over time. Our methodology is designed to be resource-efficient, making it suitable for implementation in embedded systems. We demonstrate the effectiveness of our approach using the CHB-MIT dataset, achieving an improvement of 35.34% in the F1 score with respect to a fine-tuning approach that does not consider catastrophic forgetting. Furthermore, we show that a small 1-hour data replay buffer suffices to achieve F1 scores comparable to that of a resource-unlimited scenario, while also decreasing the False Alarm Rate in 24 hours by 33% compared to a resource-unconstrained method.