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arXiv · 2602.15548

A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions

Tibor Kiss
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Abstract

In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation \[ f\big(f(-x)+x\big)=f\big(-f(x)\big)+f(x),\qquad x\in\mathbb{R}. \] Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski's Conjecture.

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Tibor Kiss. 2026-02-17. A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions. https://arxiv.org/abs/2602.15548

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