Restrictions of continuous functions

TL;DR

This paper investigates the conditions under which restrictions of continuous functions to certain subsets of [0, 1] exhibit properties like bounded variation or monotonicity, focusing on the size of these subsets measured by Hausdorff or Minkowski dimensions.

Contribution

It explores the existence of large subsets where continuous functions display improved regularity properties, using Hausdorff and Minkowski dimensions to define 'substantial' sets.

Findings

Existence of subsets with bounded variation for any continuous function.

Conditions for subsets where functions are monotone.

Analysis of the size of subsets in terms of fractal dimensions.

Abstract

Given a continuous real-valued function on [0, 1], and a closed subset E \subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The restriction f E will typically be "better behaved" than f . It may have bounded variation when f doesn't, it may have a better modulus of continuity than f, it may be monotone when f is not, etc. All this clearly depends on f and on E, and the questions that we discuss here are about the existence, for every f, or every f in some class, of "substantial" sets E such that f E has bounded total variation, is monotone, or satisfies a given modulus of continuity. The notion of "substantial" that we use is that of either Hausdorff or Minkowski dimensions.