#### Thesis Title

#### Graduation Year

2017

#### Date of Thesis Acceptance

Spring 5-10-2017

#### Major Department or Program

Mathematics

#### Advisor(s)

Russel A. Gordon

#### Abstract

In an elementary calculus course, we talk mostly, or exclusively, about integrating continuous, real-valued functions. Since continuous functions on closed intervals are integrable, the Fundamental Theorem of Calculus gives us a method to calculate these integrals (given that we can find an antiderivative). Furthermore, the Fundamental Theorem of Calculus states that the integral can be used to define an antiderivative of a continuous function. In this paper, we will present a method for establishing the existence of antiderivatives of continuous functions without using any integration theory. In addition, we will explore the potentially counter-intuitive topic of derivatives which are not Riemann integrable. It is easy to find a function whose derivative is unbounded, and thus not Riemann integrable; what is more surprising is that even bounded derivatives are not necessarily Riemann integrable. We will present two classes of functions, one conceived by Volterra and one by Pompeiu, which are differentiable on closed intervals, and whose derivatives are not Riemann integrable. Finally, we will develop the Henstock integral as a tool which integrates all derivatives.

#### Page Count

57

#### Subject Headings

Calculus -- Fundamental Theorem of Calculus, Bernhard Riemann (1826-1866), Entire Functions -- Integral, Integral calculus, Derivatives (Mathematics) Vito Volterra (1860-1940), Dimitrie Pompeiu (1873-1954), Henstock-Kurzweil integral, Whitman College 2017 -- Dissertation collection -- Mathematics Department

#### Permanent URL

http://hdl.handle.net/10349/072720171384

#### Document Type

Public Accessible Thesis

#### Recommended Citation

Granath, Elliot M., "Bounded Derivatives Which Are Not Riemann Integrable" (2017). *Honors Theses*. 341.

http://arminda.whitman.edu/theses/341

#### Rights Statement

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/

This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).