Semester

Spring

Date of Graduation

2022

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Krzysztof Chris Ciesielski

Committee Member

Paul Gartside

Committee Member

Dening Li

Committee Member

Adrian Tudorascu

Committee Member

Jerzy Wojciechowski

Abstract

This dissertation is a summary of the author's research work, supervised by Professor Krzysztof Ciesielski, based on three published articles in the Journal of Mathematical Analysis and Applications and one published article in the Banach Journal of Mathematical Analysis. Our work focuses on the study of paradoxical real functions regarding differentiability and generalized continuity within the foundations of real analysis. The reasons why they are paradoxical are directly connected to their definitions, which will be provided and explained in the later texts. Note that all functions discussed in the dissertation are single-variable and real valued, that is, well-defined from a proper or improper subset of the set R of real numbers into R.

The material is presented in two independent chapters. Chapter 1 consists of a study of nowhere-monotone differentiable functions to which we refer as the differentiable monsters. Almost 130 years after A. Köpcke constructed the first differentiable monster and after its many simplifications, K. Ciesielski noticed a few years ago the simplest such construction by far. This construction was a shifted difference of two arbitrary strictly increasing Pompeiu-like functions, that is, of differentiable functions with their derivatives vanishing on a dense subset of their domain. However, not every differentiable monster of bounded variation admits such a Jordan-like decomposition that possesses Pompeiu-likeness. We have first characterized differentiable monsters that can be decomposed in such a ``nice'' way as those that are a difference of two increasing differentiable functions. Secondly, as Jarník's extension theorem allows a differentiable extension to be as ``good'' as being smooth on the extended parts, we work on the other direction and make a differentiable extension as ``bad'' as being nowhere-monotone on the extended parts. Since it is an easy consequence of Darboux's theorem that a differentiable monster must be Pompeiu-like, we have shown that a typical function in a designated complete metric space, which consists of all differentiable extensions that are Pompeiu-like on the extended part, is nowhere-monotone on the extended part. On the other hand, we have also shown that the family of nowhere nowhere-monotone functions is dense in this space.

In Chapter 2, we additionally impose a set-theoretical axiom that the set R is not a union of less than continuum-many meager sets. A Darboux function is a function that satisfies the intermediate value property, so the classes of Darboux-like functions represent a group of functions that are continuous in a generalized sense. On the contrary, Sierpiński-Zygmund functions, first constructed in 1923 by W. Sierpiński and A. Zygmund, have as little of the standard continuity as possible. The algebra of subsets generated by these classes and the Sierpiński-Zygmund functions has nine atoms, that is, the smallest nonempty elements of the algebra. In this work, we have crafted a lemma that easily create examples in each of these nine atoms with transfinite induction. Note that the examples within the seven of the nine atoms were first discovered by K. Ciesielski and C.-H. Pan and had been included in a survey of Sierpiński-Zygmund functions in 2019 by K. Ciesielski and J. Seoane-Sepúlveda. As lineability of the main classes of Darboux-like functions, as well as of Sierpiński-Zygmund functions, has been intensively studied, our presented work has caused some on-going researches in the lineability of the nine smaller classes mentioned above.

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