Cantor Sets and Singular Functions

Abstract

In Chapter 7, we defined a singular function to be a nonconstant function whose derivative is zero a.e. The Cantor ternary function of Chapter 1 is an example of a continuous increasing singular function. In this chapter we consider singular functions in more detail. The study of such functions illustrates the beauty and subtlety of analysis in R¹ as well as providing important applications to other fields, in particular, harmonic analysis. We begin the chapter with a recap of the results found in Chapter 1 on the Cantor ternary set and function, adding the result by Randolph that the set of distances between points of the Cantor set fills up the unit interval, some geometric results from Hille and Tamarkin, and a discussion of derivatives of the Cantor ternary function. In Section 8.2, we introduce Hausdorff measure and Hausdorff dimension and calculate the Hausdorff dimension of the Cantor ternary set. Sections 8.3, 8.4, and 8.5 present generalized Cantor sets and Cantor-like sets and their corresponding functions which share many, but not all, of the properties of the Cantor ternary set and function. A modulus of continuity result is given for Cantor-like sets in Section 8.5. The final section of this chapter gives details of the construction of two strictly increasing singular functions.