Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$, defined as $f:x\mapsto \|x\|^2_2$.
It's evident that the pre-image $f^{-1}(1)$ becomes $(n-1)$-sphere $S^{n-1}$.
Consider a smooth $\epsilon$-perturbation $g$ of $f$ in the uniform sense:
$$\|f-g\|_{\operatorname{sup}}<\epsilon$$
I want to prove that $g^{-1}(1)$ can not be embedded in $\mathbb{R}^{n-1}$ if $\epsilon$ is a sufficiently small value.
If $g^{-1}(1)$ is a manifold, it is clear because a compact $(n-1)$-manifold can not be embedded in $\mathbb{R}^{n-1}$.
However, I am not sure that $g^{-1}(1)$ is a manifold because $1$ is not necessarily the regular value of $g$.
But it should be something very similar to $S^{n-1}$ because it divides the entire Euclidean space $\mathbb{R}^n$ into (at least) two components, $g^{-1}([0,1))\supset B_{1-\epsilon}(0) $ and $g^{-1}((1,\infty))\supset \mathbb{R}^n - B_{1+\epsilon}(0) $.
Any assistance you can offer would be greatly appreciated.