Modulus of continuity $\varphi(t)$ will be called non-lipschitz, if ${\varphi(t)}/{t}\to+\infty$ as $t\to+0$. Without mentioning it every time, it will be assumed that every modulus of continuity is non-lipschitz, strictly increasing, and concave up. Hausdorff $\varphi$-measure of set $e$ will be denoted as $H_{\varphi}(e)$.
$1^{\circ}$. For an arbitrary modulus of continuity $\varphi(t)$ we costruct set $C_{\varphi}$ of Cantor type, whose Hausdorff $\varphi$-measure is positive and finite. We will show that Cantor function $C_{\varphi}(x)$ constructed on set $C_{\varphi}$ belongs to Nikol'skii–Hölder class $H^{\varphi}$.
$2^{\circ}$. We prove that if modulus of continuity of function$f(x)$ satisfies the inequality $\omega_f(t)\le L\varphi(t)$, function $f(x)$ is differentiable almost everywhere on $[a;b]\setminus e$, where set $e$ is closed, $H_{\varphi}(e)<+\infty$, plus, $f'(x)$ is integrable on $[a;b]$, then we have an estimate
\begin{equation*}
\biggl|\int\limits_a^b f'(x)\,dx-f(x)\Bigm|_a^b\biggr|\le L H_{\varphi}(e).
\end{equation*}
We provide an example of set $C_{\varphi}$ and function $C_{\varphi}(x)$ that illustrates that this estimate is exact.
$3^{\circ}$. Let $\varphi(t)$ and $\psi(t)$ be moduli of continuity that satisfy the estimate $\psi(t)=o\bigl(\varphi(t)\bigr)$ as $t\to+0$. If a function $f(x)\in C[a;b]$ and for each point $\xi$ of closed set $e\subset[a;b]$ for all points $x$, sufficiently close to $\xi$, there exists $c_\xi>0$, so that the following inequality holds:
\begin{equation*}
|f(x)-f(\xi)|\ge c_\xi\varphi(|x-\xi|),
\end{equation*}
then $H_{\psi}(e)=0$.
Keywords: modulus of continuity, Nikol'skii–Hölder class, Cantor set, Cantor function, Newton–Leibniz formula, exceptional set.
Received April 7, 2025
Revised July 4, 2025
Accepted July 7, 2025
Dmitrii Sergeevich Telýakovskii, Cand. Sci. (Phys.-Math.), Prof., 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. Gerega A.N. Konstruktivnyye fraktaly v teorii mnozhestv [Constructive fractals in set theory]. Odessa, Publ. “Osvita Ukrainy”, 2017, 83 p. ISBN: 978-617-7366-32-3 .
3. Dolzhenko E.P. The removability of singularities of analytic functions. Uspehi Mat. Nauk, 1963, vol. 18, no. 4(112), pp. 135–142 (in Russian).
4. Rubinstein A.I. On $\omega$-lacunary series and functions of the classes $H^\omega$. Mat. Sb. (Nov. Ser.), 1964, vol. 65(107), no. 2, pp. 239–271 (in Russian).
5. Hille E., Tamarkin J.D. Remarks on a known example of a monotone continuous function. Amer. Math. Monthly, 1929, vol. 36, no. 5, pp. 255–264. https://doi.org/10.1080/00029890.1929.11986950
6. Saks S. Theory of the integral. New York, Hafner Publ. Comp., 1937, 367 p. Translated to Russian under the title Teoriya integrala, Moscow, Inostr. Liter. Publ., 1949, 496 p.
Cite this article as: D.S. Telýakovskii. On exceptional sets in the Newton–Leibniz formula. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 250–263.
