Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, first online, pp. 1-12

Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist

Martin Rmoutil

Received June 27, 2023. Published online March 15, 2024.

Abstract: For any $\alpha, \beta>0$ with $\alpha+\beta<1$ we provide a simple construction of an $\alpha$-Hölde function $f\colon[0,1]\to{\mathbb R}$ and a $\beta$-Hölder function $g\colon[0,1]\to{\mathbb R}$ such that the integral $\int_0^1 f {\rm d} g$ fails to exist even in the Kurzweil-Stieltjes sense.

Keywords: Kurzweil-Stieltjes integral; Hölder function; counterexample

Classification MSC: 26A39, 26A42, 26A16

DOI: 10.21136/CMJ.2024.0296-23

PDF available at: Springer Institute of Mathematics CAS

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Affiliations: Martin Rmoutil, Charles University, Faculty of Mathematics and Physics, Department of Mathematics Education, Sokolovská 83, 186 75 Prague 8 Karlín, Czech Republic, e-mail: rmoutil@karlin.mff.cuni.cz

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