In the arena of reinforcement learning, the concept of perpetual adaptation and improvement is encapsulated in the term continual reinforcement learning. This paper provides a thoughtful definition and exploration of the concept:

• A clear and formal definition of continual reinforcement learning, viewing agents that are in a constant state of learning.
• An establishment of a mathematical language for analyzing and understanding the lifecycle of continual learning agents.
• The proposition that continual learning agents can be seen as indulging in an implicit search process indefinitely.
• The presentation of examples that show traditional multi-task reinforcement learning as a subset of this broader definition.

Key Insights:

• Endless Learning: The method underscores the importance of creating agents that never cease to learn.
• Mathematical Framework: The introduction of a formalized way to study and design such agents.
• Agent Analysis: The potential of this work lies in its ability to help us conceive agents that are perpetually adapting.

The importance of this research extends to the foundation it sets for building more robust and adaptable AI systems, which is crucial for handling dynamically evolving environments and tasks. It provokes interesting questions on the nature of intelligence and the extent to which AI can genuinely emulate human-like learning.