Date of Graduation
Eberly College of Arts and Sciences
Krzysztof Chris Ciesielski.
Given two families of real functions F1 and F2 we consider the following question: can every real function f be represented as f = f 1 + f2, where f1 and f2 belong to F1 and F2 , respectively? This question leads to the definition of the cardinal function Add: Add( F1,F2 ) is the smallest cardinality of a family F of functions for which there is no function g in F1 such that g + F is contained in F2 . This work is devoted entirely to the study of the function Add for different pairs of families of real functions. We focus on the classes that are related to the additive properties and generalized continuity.;Chapter 2 deals with the classes related to the generalized continuity. In particular, we show that Martin's Axiom (MA) implies Add(D,SZ) is infinite and Add(SZ,D) equals to the cardinality of the set of all real numbers. SZ and D denote the families of Sierpinski-Zygmund and Darboux functions, respectively. As a corollary we obtain that the proposition: every function from R into R can be represented as a sum of Sierpinski-Zygmund and Darboux functions is independent of ZFC axioms.;Chapter 3 is devoted entirely to the classes related to the concept of additivity. We introduce the definition of Hamel functions. We say that a real function is a Hamel function if its graph is a Hamel basis for the plane. Main result of this chapter is the theorem that every real function can be represented as the pointwise sum of two Hamel functions.;In Chapter 4 we investigate the function Add for pairs of classes such that one relates to the generalized continuity and the other to the additive properties.
Plotka, Krzysztof, "Set-theoretic and algebraic properties of certain families of real functions" (2001). Graduate Theses, Dissertations, and Problem Reports. 1406.