Let $Z(f):=f^{-1}(0)$, i.e., the preimage of zero and $f:A\subset\mathbb{R}^m\to \mathbb{R}$.
Claim: if the function $f$ is $C^1$-smooth and 0 is not a critical value, then we can bound the number of connected components $\gamma$ of $Z(f)$ contained in $B_R$ (ball of radius R centered at 0) by the number of connected components $G$ of $U \backslash Z(f)$ compactly contained in $B_R$.
Indeed, all we need for that is to note that each $\gamma ⊂ B_R$ is the outer boundary of some $G$ compactly supported in $B_R$ and no two different connected components $\gamma ⊂ B_R$ of $Z(f)$ can serve as the outer boundary of the same connected component $G$ of $U\backslash Z(f)$. simultaneously.
Source: https://arxiv.org/abs/1507.02017 by Nazarov and Sodin
I understand the idea, but I don't see why and where the assumption of critical values is needed.
By the Submersion Theorem, we know that level sets are manifolds, without boundary, so they are geometrically closed (obviously topologically closed by definition if our function is continuous). Note that for the theorem we need regular points, check the hypotheses. Then, the results follows easily, as on the boundary of nodal components $f=0.$