On noncontinuous bisymmetric strictly monotone operations

TL;DR

This paper constructs examples of bisymmetric, strictly increasing binary operations on real intervals that are not continuous, challenging the assumption that such operations must be continuous, and explores conditions for automatic continuity.

Contribution

It provides the first known construction of noncontinuous bisymmetric, strictly increasing operations on real intervals using fractal sets, and characterizes when continuity is guaranteed.

Findings

Constructed noncontinuous bisymmetric, strictly increasing operations using Cantor-type sets.

Proved that certain symmetric, bisymmetric, strictly increasing operations are continuous if reflexive at two points.

Extended the construction to multivariate operations and identified conditions for automatic continuity.

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.