On noncontinuous bisymmetric strictly monotone operations

Abstract

We construct bisymmetric, strictly increasing binary operations on real intervals which are not continuous. This answers a natural question in the theory of bisymmetric and mean-type operations by showing that continuity may fail for non-reflexive operations of the form \[ F(x,y)=f^{-1}(\alpha f(x)+\beta f(y)), \] where $\alpha,\beta>0$ with $\alpha+\beta\neq1$. Our construction is based on a Cantor-type perfect set whose elements are linearly independent over a countable subfield of $\R$, which allows the generating function $f$ to map an interval bijectively onto a nowhere dense fractal-type set. As a consequence we obtain a noncontinuous associative and strictly increasing operation on an interval. We also extend the construction to the multivariate case. In the opposite direction we prove that if a symmetric bisymmetric strictly increasing operation is reflexive at two points of an interval, then it is automatically continuous on the segment between them and coincides there with a quasi-arithmetic mean.