semicontinuous
Suppose $X$ is a topological space^{}, and $f$ is a function from $X$ into the extended real numbers ${\mathbb{R}}^{*}$; $f:X\to {\mathbb{R}}^{*}$. Then:

1.
If ${f}^{1}((\alpha ,\mathrm{\infty}])=\{x\in X\mid f(x)>\alpha \}$ is an open set in $X$ for all $\alpha \in \mathbb{R}$, then $f$ is said to be lower semicontinuous.

2.
If $$ is an open set in $X$ for all $\alpha \in \mathbb{R}$, then $f$ is said to be upper semicontinuous.
In other words, $f$ is lower semicontinuous, if $f$ is continuous^{} with respect to the topology^{} for ${\mathbb{R}}^{*}$ containing $\mathrm{\varnothing}$ and open sets
$$U(\alpha )=(\alpha ,\mathrm{\infty}],\alpha \in \mathbb{R}\cup \{\mathrm{\infty}\}.$$ 
It is not difficult to see that this is a topology. For example, for a union of sets $U({\alpha}_{i})$ we have ${\cup}_{i}U({\alpha}_{i})=U(inf{\alpha}_{i})$. Obviously, this topology is much coarser^{} than the usual topology for the extended numbers. However, the sets $U(\alpha )$ can be seen as neighborhoods of infinity^{}, so in some sense, semicontinuous functions are ”continuous at infinity” (see example 3 below).
0.0.1 Examples

1.
A function $f:X\to {\mathbb{R}}^{*}$ is continuous if and only if it is lower and upper semicontinuous.

2.
Let $f$ be the characteristic function^{} of a set $\mathrm{\Omega}\subseteq X$. Then $f$ is lower (upper) semicontinuous^{} if and only if $\mathrm{\Omega}$ is open (closed). This also holds for the function that equals $\mathrm{\infty}$ in the set and $0$ outside.
It follows that the characteristic function of $\mathbb{Q}$ is not semicontinuous.

3.
On $\mathbb{R}$, the function $f(x)=1/x$ for $x\ne 0$ and $f(0)=0$, is not semicontinuous. This example illustrate how semicontinuous ”at infinity”.
0.0.2 Properties
Let $f:X\to {\mathbb{R}}^{*}$ be a function.
 1.

2.
Suppose $f$ is upper (lower) semicontinuous, $A$ is a topological space, and $\mathrm{\Psi}:A\to X$ is a homeomorphism. Then $f\circ \mathrm{\Psi}$ is upper (lower) semicontinuous.

3.
Suppose $f$ is upper (lower) semicontinuous, and $S:{\mathbb{R}}^{*}\to {\mathbb{R}}^{*}$ is a sense preserving homeomorphism. Then $S\circ f$ is upper (lower) semicontinuous.

4.
$f$ is lower semicontinuous if and only if $f$ is upper semicontinuous.
References
 1 W. Rudin, Real and complex analysis, 3rd ed., McGrawHill Inc., 1987.
 2 D.L. Cohn, Measure Theory, Birkhäuser, 1980.
Title  semicontinuous 

Canonical name  Semicontinuous1 
Date of creation  20130322 14:00:16 
Last modified on  20130322 14:00:16 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  13 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 26A15 
Defines  lower semicontinuous 
Defines  upper semicontinuous 
Defines  lower semicontinuous 
Defines  upper semicontinuous 