# 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)

## 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 Semicontinuous 2013-03-22 12:45:41 2013-03-22 12:45:41 drini (3) drini (3) 6 drini (3) Definition msc 26A15 msc 54-XX semicontinuous