Alzheimer's disease (AD) is a neurodegenerative disorder that can cause a significant impairment in physical and cognitive functions. Gait disturbances are also reported as a symptom of AD. Previous works have used Convolutional Neural Networks (CNNs) to analyze data provided by motion sensors that monitor Alzheimer's patients. However, these works have not explored continual learning algorithms that allow the CNN to configure itself as it receives new data from these sensors. This work proposes a method aimed at enabling CNNs to learn from a continuous stream of data from motion sensors without having full access to previous data. The CNN identifies the stage of AD from the analysis of data provided by motion sensors. The work includes an experimentation with data captured by accelerometers that monitored the activity of 35 Alzheimer's patients for a week in a daycare center. The CNN achieves an accuracy of 86,94%, 86,48% and 84,37% for 2, 3 and 4 experiences respectively. The proposal provides advantages to working with a continuous stream of data so that the CNN are constantly self-configuring without the intervention of a human. The work can be considered as promising and helpful in finding deep learning solutions in medical cases in which patients are constantly monitored.