Definition:Lower Limit (Topological Space)

Definition
Let $\left({S, \tau}\right)$ be a topological space.

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

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


 * $\displaystyle \liminf_{x \mathop \to x_0}\ f \left({x}\right) := \sup_{V \mathop \in \mho \left({x_0}\right)} \left\{ {\inf_{x \mathop \in V} \ f \left({x}\right)}\right\}$

where $\mho \left({x_0}\right)$ stands for the set of open neighborhoods of $x_0$.

Note
Because $x_0 \in V$ for all $V \mathop \in \mho \left({x_0}\right)$, the above definition implies


 * $\displaystyle \liminf_{x \mathop \to x_0}\ f \left({x}\right) \leq f \left({x_0}\right)$

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


 * $\displaystyle \liminf_{x \mathop \to x_0}\ f \left({x}\right) := \sup_{V \mathop \in \mho \left({x_0}\right)} \left\{ {\inf_{x \mathop \in V \setminus \left\{{x_0}\right\}} \ f \left({x}\right)}\right\}$

These definitions differ only if $f(x)$ 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.