Semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity.

is upper (respectively, lower) semicontinuous at a point

if, roughly speaking, the function values for arguments near

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

, then the result is upper semicontinuous; if we decrease its value to

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

is a function with values in the extended real numbers

is called upper semicontinuous at a point

where lim sup is the limit superior of the function

is a metric space with distance function

formulation, similar to the definition of continuous function.

is called upper semicontinuous if it satisfies any of the following equivalent conditions:[2] A function

is called lower semicontinuous at a point

is a metric space with distance function

is called lower semicontinuous if it satisfies any of the following equivalent conditions: Consider the function

which returns the greatest integer less than or equal to a given real number

Upper and lower semicontinuity bear no relation to continuity from the left or from the right for functions of a real variable.

Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.

while the function limits from the left or right at zero do not even exist.

[5] As an example, consider approximating the unit square diagonal by a staircase from below.

denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to

Then by Fatou's lemma the integral, seen as an operator from

For set-valued functions, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower hemicontinuity.

2.2 Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing

in the above definitions with arbitrary topological spaces.

[11] Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty.

An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map.

is upper semicontinuous in the single-valued sense but the set-valued map

is not upper semicontinuous in the set-valued sense.