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

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

First available in Project Euclid: 4 November 2011

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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

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


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.

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  • P. C. Allaart and K. Kawamura, Extreme values of some continuous nowhere differentiable functions, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 269–295.
  • L. Berg and M. Krüppel, De Rham's singular function and related functions, Z. Anal. Anwendungen 19 (2000), no. 1, 227–237.
  • G. de Rham, Sur quelques courbes definies par des equations fonctionnelles, Univ. e Politec. Torino. Rend. Sem. Mat. 16 (1957), 101–113.
  • G. A. Edgar, Classics on Fractals, Addison-Wesley, Reading, MA, 1993.
  • M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math. 1 (1984), no. 1, 183–199.
  • K. Kawamura, On the classification of self-similar sets determined by two contractions on the plane, J. Math. Kyoto Univ. 42 (2002), no. 2, 255–286.
  • M. Krüppel, De Rham's singular function, its partial derivatives with respect to the parameter and binary digital sums, Rostock. Math. Kolloq. No. 64 (2009), 57–74.
  • Z. Lomnicki and S. Ulam, Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités I. Variables indépendantes, Fund. Math. 23 (1934), 237–278.
  • T. Okada, T. Sekiguchi and Y. Shiota, An explicit formula of the exponential sums of digital sums, Japan J. Indust. Appl. Math. 12 (1995), no. 3, 425–438.
  • H. Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, Sūgaku 61 (2009), no. 2, 133–161.
  • T. Takagi, A simple example of the continuous function without derivative, Phys.-Math. Soc. Japan 1 (1903), 176–177. The Collected Papers of Teiji Takagi, S. Kuroda, Ed., Iwanami, Tokyo, 1973, 5–6.
  • H. Takayasu, Physical models of fractal functions, Japan J. Appl. Math. 1 (1984), no. 1, 201–205.
  • S. Tasaki, I. Antoniou and Z. Suchanecki, Deterministic diffusion, de Rham equation and fractal eigenvectors, Phys. Lett. A 179 (1993), no. 2, 97–102.