Continuous diagnosis and prognosis are essential for intensive care patients. It can provide more opportunities for timely treatment and rational resource allocation, especially for sepsis, a main cause of death in ICU, and COVID-19, a new worldwide epidemic. Although deep learning methods have shown their great superiority in many medical tasks, they tend to catastrophically forget, over fit, and get results too late when performing diagnosis and prognosis in the continuous mode. In this work, we summarized the three requirements of this task, proposed a new concept, continuous classification of time series (CCTS), and designed a novel model training method, restricted update strategy of neural networks (RU). In the context of continuous prognosis, our method outperformed all baselines and achieved the average accuracy of 90%, 97%, and 85% on sepsis prognosis, COVID-19 mortality prediction, and eight diseases classification. Superiorly, our method can also endow deep learning with interpretability, having the potential to explore disease mechanisms and provide a new horizon for medical research. We have achieved disease staging for sepsis and COVID-19, discovering four stages and three stages with their typical biomarkers respectively. Further, our method is a data-agnostic and model-agnostic plug-in, it can be used to continuously prognose other diseases with staging and even implement CCTS in other fields.