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November 2011 On the set of points where Lebesgue’s singular function has the derivative zero
Kiko Kawamura
Proc. Japan Acad. Ser. A Math. Sci. 87(9): 162-166 (November 2011). DOI: 10.3792/pjaa.87.162

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.

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Kiko Kawamura. "On the set of points where Lebesgue’s singular function has the derivative zero." Proc. Japan Acad. Ser. A Math. Sci. 87 (9) 162 - 166, November 2011. https://doi.org/10.3792/pjaa.87.162

Information

Published: November 2011
First available in Project Euclid: 4 November 2011

zbMATH: 1236.26007
MathSciNet: MR2863359
Digital Object Identifier: 10.3792/pjaa.87.162

Subjects:
Primary: 26A27
Secondary: 26A15 , 26A30 , 60G50

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

Rights: Copyright © 2011 The Japan Academy

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Vol.87 • No. 9 • November 2011
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