Let $f \colon \mathbb{R}^n \to \mathbb{R}$ and $h \colon \mathbb{R}^n \to \mathbb{R}^m$ be twice continuously differentiable maps with $n > m$, where $0$ is a regular value of $h$. By Sard's theorem, we see that the set of critical values of $\nabla f$ (continuously differentiable) is negligible.
Consider now the $\mathcal C^2$-manifold, $$ M = \{ x \in \mathbb{R}^n, ~h(x)=0\}. $$ We can consider the twice continuously differentiable restriction $f_{|M} \colon M \to \mathbb{R}$. We could then say that, $$ \nabla f_{|M} \colon M \to TM, $$ is continuously differentiable and proceed to apply Sard's theorem. But the map I am really interested in is not exactly $\nabla f_{|M}$ or at least not written as such, specifically I would like to only "select" the value of $\nabla f_{|M}$, and discard the point at which it applies. Any idea what I am missing here?
I think I could cover the manifold with a countable atlas, and using each chart to locally look at $f_{|M}$ as an application from $\mathbb{R}^{n-m} $ to $\mathbb{R}$, I can apply Sard's theorem as before. This tells me that for each open of my atlas, the set of critical values is negligible, then since there are only countably many negligible sets, the set of critical values is negligible.
How does that sound like?