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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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

Comments:

1. It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on every level set $E_t$. Note that this is not a contradiction!

2. The desired result is true once $f$ is $C^{n+1}$ by Sard's theorem.

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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

Comments:

1. It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on every level set $E_t$. Note that this is not a contradiction!

2. The desired result is true for $f \in C^n (\Omega)$ by Sard's theorem.

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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

Comments: It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on every level set $E_t$. Note that this is not a contradiction!

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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

Comments:

1. It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on every level set $E_t$. Note that this is not a contradiction!

2. The desired result is true once $f$ is $C^{n+1}$ by Sard's theorem.

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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

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

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 $E_t := \{x \in \Omega \, | \, f(x) = t\}$.

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 $x \in E_t$?

Comments: It is interesting to contrast the desired result to the fact that we have $Df = 0$, $\mathcal H^n$-a.e. on every level set $E_t$. Note that this is not a contradiction!