Knot points of typical continuous functions

TL;DR

This paper characterizes the sets of points where most continuous functions are nondifferentiable, using topological and game-theoretic methods to understand the typical behavior of continuous functions.

Contribution

It provides a complete characterization of families of sets of points related to the differentiability properties of most continuous functions, employing a topological zero-one law and the Banach-Mazur game.

Findings

Most continuous functions are nowhere differentiable.

The paper characterizes sets of points where nondifferentiability occurs.

Uses topological zero-one law and Banach-Mazur game in proofs.

Abstract

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.