  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.
