If $u:\mathbb{R}^n\rightarrow\mathbb{R}$ is $C^1$, then for almost every $t\in\mathbb{R}$, $\nabla u(x)\neq0$ for all $x\in\{u=t\}$.

Question

Let $u:\mathbb{R}^n\rightarrow\mathbb{R}$ be a $C^1$ function. I would like to prove that $\{u\geq t\}$ is an $n$-regular surface with boundary for a.e. $t$ using the following fact: for almost every $t\in\mathbb{R}$, $\nabla u(x)\neq0$ for all $x\in\{u=t\}$. Is this true? Any idea?

Answer 1

This is an instance of Sard's Theorem.