On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Klymchuk, S. O. Pratsiovytyi, M. V. |
| author_facet | Klymchuk, S. O. Pratsiovytyi, M. V. |
| contents | We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=δ_{α_1(x)1}+\sum^{\infty}_{k=2}\left[δ_{α_k(x)k}\prod^{k-1}_{j=1}g_{α_j (x)j}\right]\equivΔ^{G^*_3}_{α_1α_2\ldotsα_k\ldots}, $$ where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $β_{0k}=0$, $β_{1k}=q_{0k}$, $β_{2k}=q_{0k}+q_{1k}$; $(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $δ_{0k}=0$, $δ_{1k}=g_{0k}$, $δ_{2k}=g_{0k}+g_{1k}$, $k\in N$.
We found criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We pay attention to properties of level sets of the functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07257 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions Klymchuk, S. O. Pratsiovytyi, M. V. Classical Analysis and ODEs Functional Analysis Number Theory 11K55, 26A27, 26A30 We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=δ_{α_1(x)1}+\sum^{\infty}_{k=2}\left[δ_{α_k(x)k}\prod^{k-1}_{j=1}g_{α_j (x)j}\right]\equivΔ^{G^*_3}_{α_1α_2\ldotsα_k\ldots}, $$ where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $β_{0k}=0$, $β_{1k}=q_{0k}$, $β_{2k}=q_{0k}+q_{1k}$; $(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $δ_{0k}=0$, $δ_{1k}=g_{0k}$, $δ_{2k}=g_{0k}+g_{1k}$, $k\in N$. We found criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We pay attention to properties of level sets of the functions. |
| title | On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions |
| topic | Classical Analysis and ODEs Functional Analysis Number Theory 11K55, 26A27, 26A30 |
| url | https://arxiv.org/abs/2603.07257 |