semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function $f$ is upper (respectively, lower) semicontinuous at a point $x_0$ if, roughly speaking, the function values for arguments near $x_0$ are not much higher (respectively, lower) than $f\left(x_0\right).$ Briefly, a function on a domain $X$ is lower semi-continuous if its epigraph $\$ is closed in $X\times\R$, and upper semi-continuous if $-f$ is lower semi-continuous.

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point $x_0$ to $f\left(x_0\right) + c$ for some $c>0$, then the result is upper semicontinuous; if we decrease its value to $f\left(x_0\right) - c$ then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.

Definitions

Assume throughout that $X$ is a topological space and $f:X\to\overline$ is a function with values in the extended real numbers \overline=\R \cup \ = \infty,\infty/math>.

Upper semicontinuity

A function $f:X\to\overline$ is called upper semicontinuous at a point $x_0 \in X$ if for every real $y > f\left(x_0\right)$ there exists a neighborhood $U$ of $x_0$ such that f(x) for all $x\in U$.Stromberg, p. 132, Exercise 4
Equivalently, $f$ is upper semicontinuous at $x_0$ if and only if
$\limsup_ f(x) \leq f(x_0)$
where lim sup is the limit superior of the function $f$ at the point $x_0.$

If $X$ is a metric space with distance function $d$ and $f(x_0)\in\R,$ this can also be restated using an $\varepsilon$-$\delta$ formulation, similar to the definition of continuous function. Namely, for each $\varepsilon>0$ there is a $\delta>0$ such that f(x) whenever $d(x,x_0)<\delta.$

A function $f:X\to\overline$ is called upper semicontinuous if it satisfies any of the following equivalent conditions:
:(1) The function is upper semicontinuous at every point of its domain.
:(2) For each $y\in\R$, the set $f^($open in $X$, where $[ -\infty ,y)=\$.
:(3) For each $y\in\R$, the $y$-superlevel set $f^([y, \infty)) = \$ is closed in $X$.
:(4) The hypograph (mathematics), hypograph $\$ is closed in $X\times\R$.
:(5) The function $f$ is continuous when the codomain $\overline$ is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals neighborhood