Quasi-arithmetic means form one of the classical and most natural classes of means.
They are generated by continuous and strictly monotone functions, and they are closely related to the notion of bisymmetry.
A fundamental result of Aczél shows that, under suitable assumptions, bisymmetry, continuous, and strict monotonicity imply that a mean is exactly quasi-arithmetic.
In this talk, I will discuss the role of regularity assumptions in this characterization problem.
First, I will present joint results with Pál Burai and Patricia Szokol concerning the extent to which continuity can be weakened or derived from the algebraic properties of the mean.
Then I will turn to the general question of whether continuity can be omitted altogether, which leads to a rather intriguing fractal-type structure.
Finally, I will mention some related multivariable results and several open problems connected to the characterization of quasi-arithmetic means and the possible elimination of regularity assumptions.