Cutting sets of continuous functions on the unit interval

Abstract

For a function $f:[0,1]\to \mathbb{R}$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f:[0,1]\to \mathbb{R}$ 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 $F=E(f)$.