level set

def

level set is set of all points for which $f$ has some given constant value

sublevel set

sublevel set is $L_{\gamma }=\{x\in S:f(x)\le \gamma \}$

Def

If a continue function $f$ on $S\subseteq {\mathbb{R}}^n$ has nonempty sublevel set and the set is closed and bounded, then $f$ has global minimum on $S$.

Def

if $S$ is unbounded, then $f$ is coercive on $S$ if and only if all of it’s sublevel set ate bounded.