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

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.