N-ary quasi-arithmetic means and families without regularity
The classical theorems of Kolmogorov--Nagumo--de Finetti and of Aczel--Maksa characterize quasi-arithmetic means from two complementary directions: the former for compatible families of means satisfying the replacement axiom, and the latter for bisymmetric means of fixed arity. We refine both representation results by showing that the required continuity follows automatically. Our main result states that every reflexive, symmetric, bisymmetric and partially strictly increasing $n$-variable operation on a real interval is continuous and hence quasi-arithmetic. The proof is based on a recursive construction on $n$-adic rationals given by bisymmetry, and a dense-domain continuity argument. The same method also yields the regularity-free Kolmogorov--Nagumo--de Finetti theorem for compatible families of strictly increasing symmetric means.