Semester

Spring

Date of Graduation

2022

Document Type

Thesis

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Krzysztof Chris Ciesielski

Committee Member

Ela Celikbas

Committee Member

Juan Seoane-Sepu'lveda

Committee Member

Adrian Tudorascu

Committee Member

Jerzy Wojciechowski

Abstract

Consider an arbitrary $\mathcal F\subset\mathbb R^\mathbb R$, where the family $\mathbb R^\mathbb R$ of all functions from $\mathbb R$ to $\mathbb R$ is considered as a linear space over $\mathbb R$. Does $\mathcal F\cup\{0\}$ contain a non-trivial lineal subspace? If so, how big the vector space can be? These questions are at the core of lineability theory. In particular, we say that a family $\mathcal F\subset\mathbb R^\mathbb R$ is {\em lineable (in $\mathbb R^\mathbb R$)}\/ provided there exists an infinite dimensional linear space contained in $\mathcal F\cup\{0\}$.

There has been a lot of attention devoted to lineability problem of subsets of linear space of functions. For instance, the families of continuous nowhere differentiable functions and of differentiable nowhere monotone functions are lineable. It has also been known for a while that the class $D\subset\mathbb R^\mathbb R$ of Darboux functions (i.e., functions that satisfy the intermediate value property) is lineable. In fact, $D$ is $2^\mathfrak c$-lineable, that is, $D\cup\{0\}$ contains a subspace of dimension $2^\mathfrak c$, where $2^\mathfrak c$ is the cardinality of $\mathbb R^\mathbb R$. The goal of this work is to study the lineabilitiy of the subclasses of $D$ that are in the algebra generated by $D$ and seven of its subclasses (known as Darboux-like functions): extendable ($Ext$), almost continuous ($AC$), connectivity ($Conn$), peripherally continuous ($PC$), having perfect road ($PR$), having Cantor Intermediate Value Property ($CIVP$), and having Strong Cantor Intermediate Value Property ($SCIVP$).

This dissertation is arranged as follows. Chapter~1 focuses on presenting notations, definitions, and summary of all results contained in this work. In chapter~2, we give a general method to have $\mathfrak c$-lineable for all Darboux-like maps and even their restriction to Baire 2 class functions. In chapter~3, we will build some tools that allow us to show $2^\mathfrak c$-lineability (i.e., maximal lineability) for all Darboux-like subclasses of $(PC\setminus D)\cup(AC\setminus Ext)$ in the algebra. In chapter~4, we are going to construct algebraically independent sets that shall be used to achieve the maximal lineability for all Darboux-like subclasses of $D\setminus Conn.$ In the last part, in chapter~5, we will make some remarks on lineability and offer new possibilities for open problems.

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