Characterization of quasi-arithmetic means without regularity condition

TL;DR

This paper demonstrates that bisymmetry, an algebraic property, ensures the continuity of certain functions, leading to a refined characterization of quasi-arithmetic means beyond classical results.

Contribution

It shows that bisymmetry implies continuity for a broad class of functions, improving the understanding of quasi-arithmetic means without regularity assumptions.

Findings

Bisymmetry ensures continuity of certain functions.

Provides a finer characterization of quasi-arithmetic means.

Extends classical results by removing regularity conditions.

Abstract

In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function $F:I^2\to I$ is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Acz\'el, Kolmogoroff, Nagumo and de Finetti.