## Graduate Theses, Dissertations, and Problem Reports

1998

#### Document Type

Dissertation/Thesis

#### Abstract

Let {dollar}F\\subseteq\\IR\\sp\\IR.{dollar} The additivity A(F) of F is the minimum cardinality of a family {dollar}G\\subseteq\\IR\\sp\\IR{dollar} with the property that {dollar}h\\ +\\ G\\subseteq F{dollar} for no {dollar}h\\in\\IR\\sp\\IR.{dollar} The additivities of various families of functions which generalize some aspect of continuity are known. These families, referred to as the Darboux-like functions, include the almost continuous, Darboux, connectivity, perfect road, peripherally continuous, and extendable functions. The additivities of the complements of the Darboux-like families are calculated or estimated. If {dollar}F\\subseteq\\IR\\sp\\IR{dollar} the additivity of {dollar}\\IR\\sp\\IR\\\\ F{dollar} is the minimum cardinality of a family {dollar}G\\subseteq\\IR\\sp\\IR{dollar} with the property that {dollar}(h + G)\\cap F\ot=\\emptyset{dollar} for every {dollar}h\\in\\IR\\sp\\IR.{dollar} The notion of additivity leads to the definition of super-additivity which we will denote by A*. If {dollar}{lcub}\\cal F{rcub}\\subseteq\\IR\\sp\\IR{dollar} then A*{dollar}({lcub}\\cal F{rcub}){dollar} is the minimum cardinality of a family of functions G with the property that for any {dollar}H\\subseteq\\IR\\sp\\IR{dollar} such that {dollar}\\vert H\\vert < {lcub}\\rm A{rcub}(F){dollar} there is a {dollar}g\\in G{dollar} such that {dollar}g + H\\subseteq F{dollar}. The super-additivities of the Darboux-like families in {dollar}\\IR\\sp\\IR{dollar} and their complements are calculated or estimated. The Darboux-like families of {dollar}(\\IR\\sp{lcub}m{rcub})\\sp{lcub}\\IR\\sp{lcub}n{rcub}{rcub}{dollar} are considered. To generalize the results obtained for these families in {dollar}\\IR\\sp\\IR{dollar} we define (n,k)-additivity. The (n,k)-additivities of the Darboux-like families are calculated for {dollar}(\\IR\\sp{lcub}m{rcub})\\sp{lcub}\\IR\\sp{lcub}n{rcub}{rcub}{dollar}. Let {dollar}B\\sb1{dollar} stand for the collection of Baire class 1 functions in {dollar}\\IR\\sp\\IR{dollar} and D denote the Darboux functions in {dollar}\\IR\\sp\\IR.{dollar} It is known that for any finite collection {dollar}G\\subseteq B\\sb1{dollar} there is an {dollar}h\\in D\\cap B\\sb1{dollar} such that {dollar}h + G\\subseteq D.{dollar} We find the smallest cardinality of a set {dollar}H\\subseteq D\\cap B\\sb1{dollar} such that for every finite collection {dollar}G\\subseteq B\\sb1{dollar} there is an {dollar}h\\in H{dollar} such that {dollar}h\\ +\\ G\\subseteq D.{dollar} This cardinal is shown to be the cofinality of the ideal of meager sets in {dollar}\\IR.{dollar} Similiar results are obtained for cliquish and quasi-continuous functions.

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