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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.26699 |
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| _version_ | 1866913072945299456 |
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| author | Kiss, Tibor Koroknai, Dóra |
| author_facet | Kiss, Tibor Koroknai, Dóra |
| contents | In 2014, Michal Lewicki and Andrzej Olbryś proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called non-symmetrically $t$-affine, then it is $t$-affine. That is, they concluded that $f$ must fulfill the above equality without any restriction on $x$ and $y$.
In the current study, first we show that the above conditional equation implies that the function in question is locally $t$-affine. Then we derive $t$-affinity on open intervals. Finally, we formulate our main result, which generalizes the theorem of Lewicki and Olbryś for any subinterval of $\mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26699 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-symmetrically $t$-affine functions revisited Kiss, Tibor Koroknai, Dóra Classical Analysis and ODEs In 2014, Michal Lewicki and Andrzej Olbryś proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called non-symmetrically $t$-affine, then it is $t$-affine. That is, they concluded that $f$ must fulfill the above equality without any restriction on $x$ and $y$. In the current study, first we show that the above conditional equation implies that the function in question is locally $t$-affine. Then we derive $t$-affinity on open intervals. Finally, we formulate our main result, which generalizes the theorem of Lewicki and Olbryś for any subinterval of $\mathbb{R}$. |
| title | Non-symmetrically $t$-affine functions revisited |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.26699 |