Cutting sets of continuous functions on the unit interval

TL;DR

This paper characterizes the sets where continuous functions cross zero, showing that any closed nowhere dense set with certain accumulation properties can be realized as the zero-crossing set of a smooth function.

Contribution

It provides necessary and sufficient conditions for a set to be the zero-crossing set of a continuous or smooth function, generalizing previous work and constructing functions with prescribed zero-crossing sets.

Findings

E(f) is a closed nowhere dense subset of the zero set of f.

Any closed nowhere dense set with accumulation points at 0 and 1 can be realized as E(f) for some smooth f.

The paper extends previous results by characterizing zero-crossing sets for continuous and smooth functions.

Abstract

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent to the conjunction $f\in C[0,1]$ (or $f\in C^\infty [0,1]$) and $D\subset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[\{ 0\}]$ where each $x\in \{0,1\}\cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $F\subset [0,1]$ with each $x\in \{0,1\}\cap E(f)$ being an accumulation point of $F$, there exists $f\in C^\infty [0,1]$ such that $F=E(f)$.