Continuous functions with impermeable graphs

TL;DR

This paper constructs continuous functions with special impermeable graphs that intersect many functions of bounded variation, revealing complex interactions between continuity, variation, and geometric properties.

Contribution

It introduces the concept of impermeable graphs, constructs functions with such graphs, and explores their implications for Lipschitz continuity and function intersections.

Findings

Existence of H"older continuous functions with impermeable graphs.

Construction of functions with permeable and impermeable graphs.

Demonstration of functions intersecting many bounded variation functions in large sets.

Abstract

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with this property has impermeable graph, and we present further examples of functions both with permeable and impermeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a H\"older continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.