# Definition:Lower Level Set

## Definition

Let $f: S \to \overline \R$ be an extended real-valued function.

Let $\alpha \in \R$.

The $\alpha$-lower level set of $f$ is the set:

$\displaystyle \operatorname{lev} \limits_{\mathop \le \alpha} f := \left\{ {x \in S: f \left({x}\right) \le \alpha}\right\}$