Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.

Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$.

We consider for each $t \in \mathbb R$ the level set $E_t := \{x \in \Omega \, | \, f(x) = t\}$, and denote by $\partial E_t$ it’s topological boundary.

Question: Is it true that for Lebesgue almost every $t \in \mathbb R$, we have $Df(x) \neq 0$ for $\mathcal H^{n-1}$-almost every $x \in \partial E_t$?

Comments: It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on almost every level set $E_t$.