Artificial intelligence (AI) has emerged as a promising tool for predicting COVID-19 from medical images. In this paper, we propose a novel continual learning-based approach and present the design and implementation of a mobile application for screening COVID-19. Our approach demonstrates the ability to adapt to evolving datasets, including data collected from different locations or hospitals, varying virus strains, and diverse clinical presentations, without retraining from scratch. We have evaluated state-of-the-art continual learning methods for detecting COVID-19 from chest X-rays and selected the best-performing model for our mobile app. We evaluated various deep learning architectures to select the best-performing one as a foundation model for continual learning. Both regularization and memory-based methods for continual learning were tested, using different memory sizes to develop the optimal continual learning model for our app. DenseNet161 emerged as the best foundation model with 96.87\% accuracy, and Learning without Forgetting (LwF) was the top continual learning method with an overall performance of 71.99\%. The mobile app design considers both patient and doctor perspectives. It incorporates the continual learning DenseNet161 LwF model on a cloud server, enabling the model to learn from new instances of chest X-rays and their classifications as they are submitted. The app is designed, implemented, and evaluated to ensure it provides an efficient tool for COVID-19 screening. The app is available to download from https://github.com/DannyFGitHub/COVID-19PneumoCheckApp.