Measurable solutions of an alternative functional equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912551492648960 |
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| author | Tóth, Péter |
| author_facet | Tóth, Péter |
| contents | In this paper we investigate the functional equation \[ φ\left( \frac{x+y}{2} \right) \left( ψ_1(x) - ψ_2(y) \right) = 0 \hspace{20mm} \left( \mbox{ for all } x \in I_1 \mbox{ and } y \in I_2 \right) \] where $ I_1 \,, I_2 $ are open intervals of $ \mathbb{R} $, $ J = \frac{1}{2} \left( I_1 + I_2 \right) $ moreover $ ψ_1 : I_1 \rightarrow \mathbb{R} $, $ ψ_2 : I_2 \rightarrow \mathbb{R} $ and $ φ: J \rightarrow \mathbb{R} $ are unknown functions. We describe the structure of the possible solutions assuming that $ φ$ is measurable. In the case when $ φ$ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measurable solutions of an alternative functional equation Tóth, Péter Classical Analysis and ODEs 39B22, 26A15 In this paper we investigate the functional equation \[ φ\left( \frac{x+y}{2} \right) \left( ψ_1(x) - ψ_2(y) \right) = 0 \hspace{20mm} \left( \mbox{ for all } x \in I_1 \mbox{ and } y \in I_2 \right) \] where $ I_1 \,, I_2 $ are open intervals of $ \mathbb{R} $, $ J = \frac{1}{2} \left( I_1 + I_2 \right) $ moreover $ ψ_1 : I_1 \rightarrow \mathbb{R} $, $ ψ_2 : I_2 \rightarrow \mathbb{R} $ and $ φ: J \rightarrow \mathbb{R} $ are unknown functions. We describe the structure of the possible solutions assuming that $ φ$ is measurable. In the case when $ φ$ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations. |
| title | Measurable solutions of an alternative functional equation |
| topic | Classical Analysis and ODEs 39B22, 26A15 |
| url | https://arxiv.org/abs/2508.17118 |