There exist positive numbers $C$ and $c$ such that, for an arbitrary concave function $\omega(t)$ of the modulus of continuity type with $\omega(t)/t\to+\infty$ as $t\to+0$, one can construct an example of a continuous nowhere differentiable Weierstrass-type function $W_\omega(x)$ satisfying the following conditions:
$1^{\circ}$. The modulus of continuity of $W_\omega(x)$ does not exceed $C\omega(t)$.
$2^{\circ}$. For each point $x_0$, there exists a sequence $\{x_n\}$ convergent to $x_0$ and such that $|W_\omega(x_n)-W_\omega(x_0)|>c\,\omega(|x_n-x_0|)$ for each $n$.
$3^{\circ}$. At each point $x_0$, the derivative numbers of $W_\omega(x)$ take all values from the interval $[-\infty;+\infty]$.
Keywords: modulus of continuity, nowhere differentiable continuous function, derivative numbers, Weierstrass-type nowhere differentiable continuous function
Received August 7, 2024
Revised November 7, 2024
Accepted November, 18, 2024
Aleksandr Iosifovich Rubinshtein, Dr. Phys.-Math. Sci., Prof., National Research Nuclear University MEPhI, Moscow, 115409 Russia; ORCID 0000-0001-8863-5438, e-mail: rubinshtein_aleksandr@mail.ru
Dmitrii Sergeevich Telýakovskii, Cand. Sci. (Phys.-Math.), National Research Nuclear University MEPhI, Moscow, 115409 Russia; ORCID 0000-0003-1579-2154, e-mail: dtelyakov@mail.ru
REFERENCES
1. Efimov A.V. Linear methods of approximating continuous periodic functions. Mat. Sb. (Nov. Ser.), 1961, vol. 54(96), no. 1, pp. 51–90 (in Russian).
2. Bolzano B. Functionenlehre. Handwriting 1830. Königliche böhmische Gesellschaft der Wissenschaften, 1930, 207 p.
3. Mishura Y., Schied A. On (signed) Takagi — Landsberg functions: $p^th$ variation, maximum, and modulus of continuity. J. Math. Anal. Appl., 2019, vol. 473, no. 1, pp. 258–272. doi: 10.48550/arXiv.1806.05702
4. Rubinstein A.I. On $\omega$-lacunary series and funcions of the classes $H^\omega$. Mat. Sb. (Nov. Ser.), 1964, vol. 65(107), no. 2, pp. 239–271.
5. Weierstrass K. Über continuirliche functionen eines reellen arguments, die für keinen werth des letzeren einen bestimmten differentialquotienten besitzen. In: Ausgewahlte Kapitel aus der Funktionenlehre. Wiesbaden, Vieweg+Teubner Verlag, pp. 190–193. doi: 10.1007/978-3-322-91273-2_5
6. Rubinstein A.I., Telyakovskii D.S. On functions of van der Waerden type. Izv. Saratov Univ. Mathematics. Mechanics. Informatics, 2023, vol. 23, no. 3, pp. 339–347 (in Russian). doi: 10.18500/1816-9791-2023-23-3-339-347
7. Myshkis A.D. Further remarks about the problem of N. N Luzin. Uspehi Mat. Nauk (Nov. Ser.), 1957, vol. 12, no. 2(74), pp. 155–157 (in Russian).
8. Telyakovskii D.S. On monogeneity conditions. Modern problems of function theory and applications. In: Proc. 21st Inter. Saratov Winter Workshop “Modern problems of function theory and their applications”, Saratov, 2022, vol. 21, pp. 289–293 (in Russian).
9. Telyakovskii D.S. Example of a continuous nowhere-differentiable function with the modulus of continuity not exceeding a given value. Vestnik Natsional’nogo Issledovatel’skogo Yadernogo Un-ta “MIFI”, 2022, vol. 11, no. 3, pp. 228–234 (in Russian). doi: 10.56304/S2304487X22030117
10. Trokhimchuk Yu.Yu. On two problems of N. N. Luzin. Uspehi Mat. Nauk (Nov. Ser.), 1956, vol. 11, no. 5(71), pp. 215–222 (in Russian).
11. Dolzhenko E.P. On the derivative numbers of complex functions. Izv. Akad. Nauk SSSR. Ser. Mat., 1962, vol. 26, pp. 347–360 (in Russian).
Cite this article as: A.I. Rubinstein, D.S. Telýakovskii. Moscow. One example of a continuous nowhere differentiable function whose modulus of continuity does not exceed a given one. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 224–233.
