Anticipated Date of Graduation

Summer 2024

Document Type

Thesis

Degree Name

Master of Science in Mathematical Sciences

Department

Mathematical Sciences

First Advisor

John Whitaker

Abstract

The task of finding functions which are continuous but nowhere differentiable has mystified and challenged mathematicians for the past three centuries. Bernard Bolzano is believed to have constructed the first example of a continuous nowhere differentiable function on an interval in 1830. Since then, several other mathematicians have constructed continuous functions which are nowhere differentiable on the entire set of real numbers or on a dense subset of the real numbers. In this paper, I will examine the work of Hermann Schwarz, Isaac Schoenberg, and Walter Rudin in this field. I will present and explain their original constructions and proofs of continuous functions which are nowhere differentiable or non-differentiable on a dense subset of their domains, and then present a generalization of their functions and proofs. Throughout this paper, we utilize the following notation: N represents the Natural Numbers R represents the Real Numbers

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