Three remarks on a question of Acz\'el

In the 49th International Symposium on Functional Equations, J. Acz\'el asked for the monotonic solutions of a certain one-parameter family of functional equations. In this short note we find that for a certain value of the parameter the monotonic solutions are the expected ones, and that for other values this is not so. Further, we find all continuous and continuously differentiable solutions to the equations.