#### Abstract

We investigate the properties of functions into the circle group. The circle group is given by the factor group R/Z. Functions into the circle group have real-valued domain and R/Z co-domain. Every real-valued function has an analogous function into the circle group. By wrapping the graph of a real-valued function around a horizontal cylinder with a circumference of one, we visualize the analogous function into the circle group. How does wrapping a real-valued function around such a cylinder affect the function outputs, limits, continuity, and rate of change? Function outputs are naturally transformed to reside on a circle with a circumference of one. Consequently, while every real-valued function has an analogous function into the circle group, this transformation is not one-to-one, i.e. two non-equal real-valued functions may have equal analogous functions into the circle group. Function limits, continuity, and rate of change are preserved with respect to this transformation. More interestingly, we find that for some real-valued functions non-existent limits become existent for the analogous functions into the circle group. Similarly, some discontinuous real-valued functions have continuous analogous functions into the circle group, and some non-differentiable real-valued functions have differentiable analogous functions into the circle group.

#### Semester/Year of Award

Fall 2020

#### Mentor

Shane P. Redmond

#### Mentor Department Affiliation

Mathematics and Statistics

#### Access Options

Open Access Thesis

#### Degree Name

Honors Scholars

#### Degree Level

Bachelors

#### Department

Mathematics and Statistics

#### Recommended Citation

Richburg, Renee O., "Going in Circles: An Exploration of Functions into the Circle Group" (2020). *Honors Theses*. 798.

https://encompass.eku.edu/honors_theses/798