Handwritten character recognition plays a vital role in digitalizing ancient scripts and manuscripts, particularly in the context of Tamil palm leaf manuscripts. This study proposes a Deep Continual Learning-based Scheme for Tamil handwritten character recognition, leveraging ResNet-based Convolutional Neural Networks (CNNs) for feature extraction and Elastic Weight Consolidation (EWC) for continual learning. The proposed method effectively mitigates catastrophic forgetting and adapts to evolving character variations while maintaining high recognition accuracy. A softmax classifier is employed for final classification, ensuring optimal differentiation among 122 distinct Tamil characters. Experimental evaluations demonstrate that the proposed model outperforms existing methodologies, achieving an accuracy of 96%. Comparative analysis with state-of-the-art techniques highlights the robustness and adaptability of the model in handling complex handwritten variations. Despite a slight trade-off in recognition speed due to the deep architecture, the proposed approach establishes a new benchmark in Tamil handwritten character recognition, offering improved reliability for real-world applications.