# Definition:Lower Semicontinuous

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## Contents

## Definition

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

Let $S$ be endowed with a topology $\tau$.

Then $f$ is **lower semicontinuous** at $\bar x \in S$ if and only if:

- $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right) = f \left({\bar x}\right)$

where $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right)$ stands for the lower limit of $f$ at $\bar x$.

### Lower Semicontinuous on Subset

Let $A \subseteq S$, and $A \ne \varnothing$.

The function $f$ is said to be **lower semicontinuous on** $A$ if and only if $f$ is lower semicontinuous at every $a \in A$.

## Also known as

When the context is clear, the abbreviation **LSC** can be used to mean **lower semicontinuous**.

## Also see