Continuous Functions that Cut the Real Axis Very Often

TL;DR

This paper explores continuous functions on [0,1] that intersect the real axis on sets of positive measure, presenting examples with unbounded variation and functions with derivatives of all orders.

Contribution

It constructs explicit examples of continuous functions that frequently cross the real axis, demonstrating both unbounded variation and infinite differentiability.

Findings

Functions can have positive measure sets where they cross zero.

Examples include functions with unbounded variation.

Some functions are infinitely differentiable.

Abstract

We consider continuous functions f : [0,1] \to R that cut the real axis at every point of a measurable set of positive measure and we construct examples where f fails to have bounded variation, and at the opposite end, where f admits derivatives of all orders.