TITLE: Does there exist a submersion $f:\mathbb{R}^{3}\setminus\{0\} \to \mathbb{R}$ with one pre-image compact and another pre-image non-compact?
QUESTION [5 upvotes]: Does there exist a submersion $f:\mathbb{R}^{3}\setminus\{0\} \to \mathbb{R}$ for which there are $c_1$ and $c_2$ in $\mathbb{R}$ such that $f^{-1}(c_1)$ is compact and $f^{-1}(c_2)$ is non-compact.

REPLY [4 votes]: Why not try something simpler, like $f=\frac{x^{2}+y^{2}+z^{2}}{1+z^{2}}$.
Then it is easy to see that $\operatorname{grad}f$ is a non-zero vector at
each point of $\mathbb{R}^{3}\backslash\{0\}$, so $f$ is a submersion; on the
other hand
the level set $f^{-1}(1):~x^{2}+y^{2}=1$ is a cylinder, hence noncompact, while
the level set $f^{-1}(\frac{1}{2}):~x^{2}+y^{2}+\frac{1}{2}z^{2}=\frac{1}{2}$
is an ellipsoid, hence compact.