Decomposition Of The Graphs Of Continuous Functions And A Class Of Cut-out Sets

In this thesis, we consider the decomposition of the continuous functions in terms of the dimension of its graph, as well as the fractal measure of a class of cut-out sets.After the introductory chapter, we present some preliminaries in the second chapter. Then we discuss these issues in details in the next three chapters.In the third chapter, we consider the relationship between the decomposition of contin-uous functions and the Hausdorff dimension. We give an affirmative answer to a question posed by Bayart and Heurtaeux. Precisely, we show that given a real β∈[1,2], any real-valued continuous function on the unit interval can be decomposed as a sum of two real-valued continuous functions whose graphs have Hausdorff dimension β.The fourth chapter is divided into two parts:in the first part, we extend the result of Humke and Petruska via relationship between packing dimension and upper box dimen-sion. We show that for a typical continuous function f∈C(X) over a compact subset X(?)Rd which contains uncountable many points, the packing dimension of its graph is dimp(X)+1; in the second part, we consider the decomposition of continuous function in terms of packing dimension. First we show that for f, g∈C(X) with dimp(Gg)≠dimp(Gf), applying it to the decomposition of the continuous functions, we show that for any f∈C([0,1]), there exist functions g,h∈C([0,1]) such that if and only if dimP(Gf)≤β. We also show that is a1-prevalent set in C(X). In the fifth chapter, we give an estimation of the h-packing measure and the h-Hausdorff measure of a class of cut-out sets, where h is a dimension function satisfying the doubling condition.In the last chapter, we summarize the main results of this dissertation, and list some topics for further research.