Properties of Pathological Functions and their Classes

Abstract

This paper is meant to serve as an exposition on the theorem proved by Richard Aron, V.I. Gurariy and J.B. Seoane in 2004 which states that ‘The Set of Differentiable but Nowhere Monotone Functions is Lineable.’ I begin by showing how certain properties that our intuition generally associates with continuous or differentiable functions can be wrong. This is followed by examples of functions with surprisingly unintuitive properties that have appeared in mathematical literature in relatively recent times. After a brief discussion of some preliminary concepts and tools needed for the rest of the paper, a construction of an ‘Continuous but Nowhere Monotone (CNM)’ function is provided which serves as a first concrete introduction to the so called “pathological functions” in this paper. The existence of such functions leaves one wondering about the ‘Differentiable but Nowhere Monotone (DNM)’ functions. Therefore, the next section offers a description of how the 2004 paper by the three authors shows that such functions are very abundant and not mere fancy counter-examples that can be called “pathological.”