semi-continuous

A real function $f:A\rightarrow\mathbbmss{R}$ , where $A\subseteq\mathbbmss{R}$ is said to be lower semi-continuous in $x_{0}$ if

\forall\varepsilon>0\ \exists\delta>0\ \forall x\in A\ |x-x_{0}|<\delta% \Rightarrow f(x)>f(x_{0})-\varepsilon,

and $f$ is said to be upper semi-continuous if

\forall\varepsilon>0\ \exists\delta>0\ \forall x\in A\ |x-x_{0}|<\delta% \Rightarrow f(x)<f(x_{0})+\varepsilon.

Remark

A real function is continuous in $x_{0}$ if and only if it is both upper and lower semicontinuous in $x_{0}$ .

We can generalize the definition to arbitrary topological spaces as follows.

Let $A$ be a topological space. $f:A\to\mathbbmss{R}$ is lower semicontinuous at $x_{0}$ if, for each $\varepsilon>0$ there is a neighborhood $U$ of $x_{0}$ such that $x\in U$ implies $f(x)>f(x_{0})-\varepsilon$ .

Theorem

Let $f:[a,b]\rightarrow\mathbbmss{R}$ be a lower (upper) semi-continuous function. Then $f$ has a minimum (maximum) in $[a,b]$ .

Title	semi-continuous
Canonical name	Semicontinuous
Date of creation	2013-03-22 12:45:41
Last modified on	2013-03-22 12:45:41
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Definition
Classification	msc 26A15
Classification	msc 54-XX
Synonym	semicontinuous