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

TL;DR

This paper disproves Matkowski's conjecture by showing the existence of continuous solutions to a specific functional equation that are not linear, and also provides conditions under which the conjecture holds.

Contribution

The paper presents a counterexample to Matkowski's conjecture, demonstrating the richness of continuous solutions and identifying conditions that enforce linearity.

Findings

Existence of continuous solutions with arbitrary parts.

Counterexample disproving the conjecture.

Sufficient conditions for the conjecture to hold.

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.