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 $[a,b]$ .