A Counterexample to Matkowski’s Conjecture for Quasi Graph-Additive Functions
Abstract
Abstract In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation $$\begin{aligned}\begin{aligned} f\big (f(-x)+x\big )=f\big (-f(x)\big )+f(x),\qquad x\in \mathbb {R}. \end{aligned}\end{aligned}$$ f ( f ( - x ) + x ) = f ( - f ( x ) ) + f ( x ) , x ∈ 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.
Fields of Study & Subjects
Functional Equations Stability Results
Mathematical Dynamics and Fractals
Meromorphic and Entire Functions
Publication details
- Journal / source
- Aequationes Mathematicae
- Publisher
- Springer Science and Business Media LLC
- Volume
- 100
- Issue
- 2
- Publication date
- 27 February 2026
- Language
- EN