The constant-breadth cam mechanisms, which drive a parallel flat-faced double follower, are desmodromic, and guarantee global bilaterality. In the said mechanisms, the law of displacement of the follower can only be freely designed for an interval of the rotation angle of the cam – the designed segment – close or equal to 180°, and the remaining interval – the calculated segment – is obtained through calculation from the first. Guaranteeing the continuity between segments is not a trivial task. This work shows a design procedure which guarantees automatically the global continuity of the law of displacement. This procedure provides the expressions of calculation for both parallel flat-faced double translating follower and parallel flat-faced double oscillating follower. Non-parametric Bézier curves are used for the definition of the displacement functions. For both types of followers, two numerical examples of the design of the displacement functions are given. Furthermore, the corresponding cam profiles are provided.