Suppose $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e. $$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$ then, for every fixed $c \in \mathbb{R}$, the level set: $$f^{-1}(c)=\{x \in \mathbb{R}^{N+1}: f(x)=c\}$$ is a $C^1$ manifold of dimension $N$.
Now suppose that not all his points are regular and let $Crit(f)$ be the set of critical points of $f$ i.e. $$Crit(f):=\{x \in \mathbb{R}^{N+1}: \nabla f (x)=0 \}$$ Suppose that, with a variant of Sard theorem, we can prove that: $$H_N \left( f^{-1}(c) \cap Crit(f) \right)=0 \; \; \; for \;a.e. \; c \in \mathbb{R}$$ where $H_N$ is the $N$-dimensional Hausdorff measure.
The question is: can I say that $f^{-1}(c)$ is a $C^1$ manifold of dimension $N$ for $a.e. \; c \in \mathbb{R}$?
Thanks.