Continuous gait design and control enable humanoid robots to smoothly transition and switch between different gaits, adapting to various task requirements, which is crucial for their real-world applications. Traditional gait control methods often rely on predefined rules and models, limiting the flexibility and adaptability of robots. To overcome the above limitations, this study combines adaptive motion functions (AMF) with reinforcement learning (RL) to achieve continuous gait design and control. Firstly, to enable a single policy to achieve different gaits, both the AMF and reward functions are designed as piecewise functions. Secondly, to enhance the flexibility of the AMF, the RL strategy is used to control the motion cycle of the AMF. This allows the robot to learn how to adjust the speed and rhythm of the gaits, achieving smooth gait transitions and switches. Lastly, to fully leverage the advantages of RL, the output of the policy is not directly summed with the AMF as the robot's action command. Instead, the policy output is adjusted and then added to the AMF, with the adjustment factor also being an output of the policy. The method proposed in this paper controls the gait cycle and adjustment factors through policies, improving the flexibility and adaptability of robots and providing insights for the practical application of continuous gaits in humanoid robots.