Definition:Lower Limit (Topological Space)

This page is about Lower Limit in the context of Topological Space. For other uses, see Lower Limit.

Definition

Let $\struct {S, \tau}$ be a topological space.

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

The lower limit of $f$ at some $x_0 \in S$ is defined as:

$\ds \liminf_{x \mathop \to x_0} \map f x := \sup_{V \mathop \in \map \mho {x_0} } \set {\inf_{x \mathop \in V} \map f x}$

where $\map \mho {x_0}$ stands for the set of open neighborhoods of $x_0$.

Note

This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: Extract this statement into its own page You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page.

Because $x_0 \in V$ for all $V \mathop \in \map \mho {x_0}$, the above definition implies:

$\ds \liminf_{x \mathop \to x_0} \map f x \le \map f {x_0}$

Also defined as

Some authors exclude $x_0$ from the infimum in the definition:

$\ds \liminf_{x \mathop \to x_0} \map f x := \sup_{V \mathop \in \map \mho {x_0} } \set {\inf_{x \mathop \in V \mathop \setminus \set {x_0} } \map f x}$

These definitions differ only if $\map f x$ is discontinuous at $x_0$.

Also see

• Definition:Limit Inferior: do not confuse that with this. The lower limit of a function is a topological property in the sense that it depends on the underlying topology of the space.

• Relationship between Limit Inferior and Lower Limit