## 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

#### 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

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

#### 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

#### References

• 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.