Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the set of points where Lebesgue’s singular function has the derivative zero

Kiko Kawamura

Full-text: Open access

Abstract

Let $L_{a}(x)$ be Lebesgue’s singular function with a real parameter $a$ ($0<a<1, a \neq 1/2$). As is well known, $L_{a}(x)$ is strictly increasing and has a derivative equal to zero almost everywhere. However, what sets of $x \in [0,1]$ actually have $L_{a}'(x)=0$ or $+\infty$? We give a partial characterization of these sets in terms of the binary expansion of $x$. As an application, we consider the differentiability of the composition of Takagi’s nowhere differentiable function and the inverse of Lebesgue’s singular function.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 9 (2011), 162-166.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1320417394

Digital Object Identifier
doi:10.3792/pjaa.87.162

Mathematical Reviews number (MathSciNet)
MR2863359

Zentralblatt MATH identifier
1236.26007

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A30: Singular functions, Cantor functions, functions with other special properties 60G50: Sums of independent random variables; random walks

Keywords
Takagi’s function Lebesgue’s singular function nowhere-differentiable function

Citation

Kawamura, Kiko. On the set of points where Lebesgue’s singular function has the derivative zero. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 162--166. doi:10.3792/pjaa.87.162. https://projecteuclid.org/euclid.pja/1320417394


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