| Continuity and differentiability are the important part of classical function, the emergence of Weierstrass function provided the basis for the continuous nondifferentiable function, and broke a new research area. More and more mathematicians are focusing on constructing some new continuous nondifferentiable functions. The most concern is Takagi function and Van Der Waerden function.The Takagi functionτ(x), a function defined on the unit interval x∈[0,1], was introduced by Takagi in 1903 as an example of a continuous nondifferentiable function. It can be defined as where ||x|| is the distance from x to the nearest integer, i.e.||x||= min{|x-n|, n∈Z}. This paper is a review paper. Through the introduction and analysis of the Takagi function, the main results are the proving of several important property theorems.The introductory chapter introduces the constructing background of the Takagi; then reviews and makes some remarks on the pioneer's work and their achievements. Chapter Two gives a detailed introduction to the Takagi function, which firstly proves that the Takagi function is a continuous nondifferentiable function and secondly presents and minutely proves the properties of the Takagi function. For example, if the Takagi function is defined in the rational number field, thenτ(x) is in the set of rational num-bers;τ(x) is the unique continuous function on [0,1] that satisfies both these functional equations:τ(x)=τ(1-x),τ(1/2x)=1/2x+1/2τ(x);Takagi exacts self-similarity, etc. Fi-nally, this chapter proves that this graph of the Takagi function has Hausdorff dimension and Box dimension 1 in R. In Chapter 3, by introducing the definition and properties of local level set, and deficient digit setΩL which comprises the set of leftmost endpoints of all local level sets, this paper obtains the properties of the generic level and its relations to the local level set. |
