A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917277945823232 |
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| author | Kiss, Tibor |
| author_facet | Kiss, Tibor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15548 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions Kiss, Tibor Classical Analysis and ODEs 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. |
| title | A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2602.15548 |