In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, semicontinuity (or semi-continuity) is a property of
extended real-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s that is weaker than
continuity. An extended real-valued function
is upper (respectively, lower) semicontinuous at a point
if, roughly speaking, the function values for arguments near
are not much higher (respectively, lower) than
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
to
for some
, then the result is upper semicontinuous; if we decrease its value to
then the result is lower semicontinuous.
![Lower semi](https://upload.wikimedia.org/wikipedia/commons/b/b0/Lower_semi.svg)
The notion of upper and lower semicontinuous function was first introduced and studied by
René Baire
René ('' born again'' or ''reborn'' in French) is a common first name in French-speaking, Spanish-speaking, and German-speaking countries. It derives from the Latin name Renatus.
René is the masculine form of the name ( Renée being the femin ...
in his thesis in 1899.
Definitions
Assume throughout that
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and
is a function with values in the
extended real number
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
s