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arXiv · 2601.16247

On noncontinuous bisymmetric strictly monotone operations

Gergely Kiss
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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 \[ F(x,y)=f^{-1}(αf(x)+βf(y)), \] where $α,β>0$ with $α+β\neq1$. 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.

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Gergely Kiss. 2026-03-06. On noncontinuous bisymmetric strictly monotone operations. https://arxiv.org/abs/2601.16247

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