Cutting sets of continuous functions on the unit interval

For a function $fcolon [0,1]tomathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $fcolon [0,1]tomathbb R$ and a Cantor set $Dsubset [0,1]$ with ${0,1}subset D$, we obtain conditions equivalent to the conjunction $fin C[0,1]$ (or $fin C^infty [0,1]$) and $Dsubset E(f)$. 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 $xin {0,1}cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $Fsubset [0,1]$ with each $xin {0,1}cap E(f)$ being an accumulation point of $F$, there exists $fin C^infty [0,1]$ such that $F=E(f)$.