Predictive machine learning models nowadays are often updated in a stateless and expensive way. The two main future trends for companies that want to build machine learning-based applications and systems are real-time inference and continual updating. Unfortunately, both trends require a mature infrastructure that is hard and costly to realize on-premise. This paper defines a novel software service and model delivery infrastructure termed Continual Learning-as-a-Service (CLaaS) to address these issues. Specifically, it embraces continual machine learning and continuous integration techniques. It provides support for model updating and validation tools for data scientists without an on-premise solution and in an efficient, stateful and easy-to-use manner. Finally, this CL model service is easy to encapsulate in any machine learning infrastructure or cloud system. This paper presents the design and implementation of a CLaaS instantiation, called LiquidBrain, evaluated in two real-world scenarios. The former is a robotic object recognition setting using the CORe50 dataset while the latter is a named category and attribute prediction using the DeepFashion-C dataset in the fashion domain. Our preliminary results suggest the usability and efficiency of the Continual Learning model services and the effectiveness of the solution in addressing real-world use-cases regardless of where the computation happens in the continuum Edge-Cloud.