Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


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Rubinstein A. I., Telyakovskii D. S. On functions of van der Waerden type. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 3, pp. 339-347. DOI: 10.18500/1816-9791-2023-23-3-339-347, EDN: BUXAKG

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Published online: 
31.08.2023
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BUXAKG

On functions of van der Waerden type

Autors: 
Rubinstein Aleksandr I., National Research Nuclear University MEPhI
Telyakovskii Dmitrii S., National Research Nuclear University MEPhI
Abstract: 

Let $\omega(t)$ be an arbitrary modulus of continuity type function, such that $\omega(t)/t\to+\infty$, as $t\to+0$. We construct a continuous nowhere-differentiable function $V_\omega(x)$, $x\in[0;1]$, that satisfies the following conditions: 1)  its modulus of continuity satisfies the estimate $O(\omega(t))$ as $t\to+0$; 2) for some positive $c$ at each point $x_0$ holds $\limsup{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}>c$ as $x\to x_0$; 3) at each point $x_0$ holds $\liminf{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}=0$ as $x\to x_0$.

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Received: 
26.04.2022
Accepted: 
04.11.2022
Published: 
31.08.2023