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

Abstract In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation $$\begin{aligned}\begin{aligned} f\big (f(-x)+x\big )=f\big (-f(x)\big )+f(x),\qquad x\in \mathbb {R}. \end{aligned}\end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.