Let $I=[0,1]^d$ be the $d$-dimensional unit box, let $f\in C^1(I)$ a real-valued function, and let $y\in \mathbb{R}$ be in the image of $f$.
My question: What will be a "miniml" sufficient condition (if possible, also neccesary), so that $m(f^{-1} (y))=0$, where $m$ is the Lebesgue measure on $I$?