La funcion de Takagi

Abstract

Takagi’s function is defined, using modern notation as τ(x) = X∞ n=0 1 2 n ϕ(2 n x),

where ϕ(x) = dist(x, Z), the distance from x to the nearest integer. This function was introduced by Teiji Takagi (1875–1960) in 1903. Takagi is famous for his work in number theory. He proved the fundamental theorem of Class Field Theory (1920, 1922). He was sent to Germany 1897-1901. He visited Berlin and Gottingen, saw Hilbert. Takagi ´ (1903) showed that the function τ(x) is continuous on [0, 1] and has no derivative at each point x ∈ [0, 1] on either side. Van der Waerden (1930) discovered the base 10 variant, proved nondifferentiability. De Rham (1956) also rediscovered the Takagi function. The Takagi function has been extensively studied in all sorts of ways, during its 100 year history, often in more general contexts. It has some surprising connections with number theory and (less surprising) with probability theory. We present below a brief summary of the content of this work. We start by studying the p-base development of a real number. In the case of binary expansions, we introduce the functions “digit sums” and “deficient digit function”, that Takagi handles them in the original definition of τ(x) In chapter 2 we present three definitions of Takagi function. For the first one, we consider the absolute value function, h(x) = |x| in the interval [−1, 1] and its periodical extension to all R. The Takagi function is g(x) = X∞ n=0 hn(x) 2 n , where hn(x) = h(2 n x). More common is to define the Takagi function using the nearest integer distance function that we denote ϕ(x), and we get the definition given at the beginning. The relationship between this two functions is τ(x) = 1 2 g(2x). Using the functions digit sum and deficient digit function we present the Takagi original definition. In this chapter we also study the maximum of the Takagi function, and we prove that the set where it reaches that maximum is perfect and of Lebesgue zero measure. We end this chapter by demonstrating some functional equations that verify the Takagi function (it is the only one that verifies certain equations) and proving that it takes rational into rational, and in particular, dyadics into dyadics. In the third chapter we study two properties of the Takagi function. On the one hand, continuity in Holder’s sense and on the other, its differentiability. We show that it is Holder continuous of class ¨ α para todo α ∈ (0, 1) and we present three demonstrations that the Takagi function does not admit a derivative (neither finite nor infinite) at any point in the interval [0, 1]. Even that it does not have a finite lateral derivative at any point of the interval [0, 1] We finish this work studying the theorem of Banach-Mazurkiewicz. For this purpose, we need Baire’s theorem, of which we give a proof. In 1929, the problem on the massiveness of the set of non-differentiable functions in the space of continuous functions was formulated by Steinhaus. In 1931, this problem was solved independently and by different ways by Banach and Mazurkiewicz. They proved that the set of non-differentiable functions in the space C([0, 1]) of functions, that are continuous on [0, 1], with the uniform metric is a set of the second category.