# Functions of Bounded Variations and Their Properties

## Abstract

Bounded variation, as a topic, was originally developed in 1881 as mathematicians were looking for criteria that would guarantee the convergence of Fourier Series. The Dirichlet-Jordan Theorem tells us that a function f that is of bounded variation is guaranteed to have a Fourier Series that converges. This theorem led mathematicians to believe that this property of functions was important, due to the relationship to Fourier Series. Thus, it grew into an interesting study in its own right, and the scope of functions of bounded variation is much broader. When considering when the variation of a function, or the total “vertical movement” of a function over an interval, there are several interesting properties to come about. This thesis explores a wide variety of properties of functions of bounded variations, as well and explores some of the ways that this class was extended to classes of generalized bounded variation. Bounded variation’s usefulness extends far beyond that of Fourier series and into several other parts of real analysis.

Pierce, Pamela

Mathematics

Analysis

## Keywords

bounded variation, real analysis, variation

2019

Bachelor of Arts

## Document Type

Senior Independent Study Thesis

COinS