An Abridged History of the Function and Mathematical Rigour

The Integrability of Thomae's Function

Abstract

For many people, the terms "mathematical function" probably call to mind something resembling the equations we are all familiar with from our primary and secondary school experiences, like F=ma, or perhaps a quadratic function. Even for many first and second year post-secondary students, polynomials, sine and cosine functions, and the exponential function probably comprise the bulk of what we consider functions. And for much of history this natural, fairly intuitive notion of functionality has proven sufficient. It was only by the beginning of the 19th century that, as the movement began to put all of mathematics on a more rigorous footing, this characterization of a function was found to be lacking. It was in this context that Dirichlet, and later Thomae, put forth the functions that bear their names as examples which demonstrated that a much broader notion of what constitutes a function could exist within the new more rigorous framework which mathematics was moving toward. That Thomae's function, a peculiar and counter-intuitive looking construction, would prove to be integrable in spite of possessing infinitely many discontinuities, was a potent demonstration of the power of the new rigour, as well as that a broad new class of mathematical objects might reasonably be termed and treated as functions, as much as the old familiar ones which preceded them. The trend toward greater mathematical rigour begun in the 19th century defines much of mathematics today, and Thomae's function serves as an important landmark which connects that past to the present.

Discipline: Mathematics

Faculty Mentor: Dr. Nataliya Zadorozhna