 
Summary: Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1
Extreme values of some continuous
nowhere differentiable functions
By PIETER C. ALLAART and KIKO KAWAMURA
University of North Texas
(Received )
Abstract
We consider the functions Tn(x) defined as the nth partial derivative of Lebesgue's
singular function La(x) with respect to a at a = 1
2 . This sequence includes a multiple
of the Takagi function as the case n = 1. We show that Tn is continuous but nowhere
differentiable for each n, and determine the H¨older order of Tn. From this, we derive
that the Hausdorff dimension of the graph of Tn is one. Using a formula of Lomnicki and
Ulam, we obtain an arithmetic expression for Tn(x) using the binary expansion of x, and
use this to find the sets of points where T2 and T3 take on their absolute maximum and
minimum values. We show that these sets are topological Cantor sets. In addition, we
characterize the sets of local maximum and minimum points of T2 and T3.
AMS 2000 subject classification: Primary 26A30,28A80; Secondary 26A27.
authors' address: Mathematics Department, P.O. Box 311430, Denton, TX 762031430,
USA; email: allaart@unt.edu, kiko@unt.edu
