Definition:(Lower) Level Set

Definition

Let $f:S\to\R\cup\left\{-\infty,\infty\right\}$ be an extended real-valued function.

Let $\alpha\in\R$.

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

$\operatorname{lev}_{\leq \alpha} f := \left\{{x \in S: f \left({x}\right) \le \alpha}\right\}$