Semi-continuity


In mathematical analysis, semicontinuity is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper 'semicontinuous' at a point if, roughly speaking, the function values for arguments near are not much higher than Briefly, a function on a domain is lower semi-continuous if its epigraph is closed in, and upper semi-continuous if 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 to for some, then the result is upper semicontinuous; if we decrease its value to 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 is a topological space and is a function with values in the extended real numbers.

Upper semicontinuity

A function is called upper semicontinuous at a point if for every real there exists a neighborhood of such that for all.
Equivalently, is upper semicontinuous at if and only if
where lim sup is the limit superior of the function at the point, defined as
where the infimum is over all neighborhoods of the point.
If is a metric space with distance function and this can also be restated using an - formulation, similar to the definition of continuous function. Namely, for each there is a such that whenever
A function is called upper semicontinuous if it satisfies any of the following equivalent conditions:

Lower semicontinuity

A function is called lower semicontinuous at a point if for every real there exists a neighborhood of such that for all.
Equivalently, is lower semicontinuous at if and only if
where is the limit inferior of the function at point
If is a metric space with distance function and this can also be restated as follows: For each there is a such that whenever
A function is called lower semicontinuous if it satisfies any of the following equivalent conditions:

Examples

Consider the function piecewise defined by:
This function is upper semicontinuous at but not lower semicontinuous.
The floor function which returns the greatest integer less than or equal to a given real number is everywhere upper semicontinuous. Similarly, the ceiling function is lower semicontinuous.
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. For example the function
is upper semicontinuous at while the function limits from the left or right at zero do not even exist.
If is a Euclidean space and is the space of curves in , then the length functional which assigns to each curve its length is lower semicontinuous. As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length.
A fundamental example in real analysis is Fatou's lemma. It asserts that if is a sequence of non-negative measurable functions, then
where denotes the limit inferior. What this means, in full generality, is that if be a measure space and denotes the set of positive measurable functions endowed with the topology of convergence in measure with respect to then the integral, seen as an operator from to is lower semicontinuous.

Properties

Unless specified otherwise, all functions below are from a topological space to the extended real numbers Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.
  • A function is continuous if and only if it is both upper and lower semicontinuous.
  • The characteristic function or indicator function of a set is upper semicontinuous if and only if is a closed set. It is lower semicontinuous if and only if is an open set.
  • In the field of convex analysis, the characteristic function of a set is defined differently, as if and if. With that definition, the characteristic function of any is lower semicontinuous, and the characteristic function of any is upper semicontinuous.

    Binary operations on semicontinuous functions

Let.
  • If and are lower semicontinuous, then the sum is lower semicontinuous. The same holds for upper semicontinuous functions.
  • If and are lower semicontinuous and non-negative, then the product function is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
  • The function is lower semicontinuous if and only if is upper semicontinuous.
  • If and are upper semicontinuous and is non-decreasing, then the composition is upper semicontinuous. On the other hand, if is not non-decreasing, then may not be upper semicontinuous. For example take defined as. Then is continuous and , which is not upper semicontinuous unless is continuous.
  • If and are lower semicontinuous, their maximum and minimum are also lower semicontinuous. Consequently, the set of all lower semicontinuous functions from to forms a lattice. The corresponding statements also hold for upper semicontinuous functions.

    Optimization of semicontinuous functions

  • The supremum of an arbitrary family of lower semicontinuous functions is lower semicontinuous.
  • If is a compact space and is upper semicontinuous, then attains a maximum on If is lower semicontinuous on it attains a minimum on

    Other properties

  • Let be a metric space. Every lower semicontinuous function is the limit of a point-wise increasing sequence of extended real-valued continuous functions on In particular, there exists a sequence of continuous functions such that
  • Any upper semicontinuous function on an arbitrary topological space is locally constant on some dense open subset of
  • If the topological space is sequential, then is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any and any sequence that converges towards, there holds. Equivalently, in a sequential space, is upper semicontinuous if and only if its superlevel sets are sequentially closed for all. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.

    Semicontinuity of set-valued functions

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.
A set-valued function from a set to a set is written For each the function defines a set
The preimage of a set under is defined as
That is, is the set that contains every point in such that is not disjoint from.

Upper and lower semicontinuity

A set-valued map is upper semicontinuous at if for every open set such that, there exists a neighborhood of such that
A set-valued map is lower semicontinuous at if for every open set such that there exists a neighborhood of such that
Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing and in the above definitions with arbitrary topological spaces.
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.
For example, the function defined by
is upper semicontinuous in the single-valued sense but the set-valued map is not upper semicontinuous in the set-valued sense.

Inner and outer semicontinuity

A set-valued function is called inner semicontinuous at if for every and every convergent sequence in such that, there exists
a sequence in such that and for all sufficiently large
A set-valued function is called outer semicontinuous at if for every convergence sequence in such that and every convergent sequence in such that for each the sequence converges to a point in .

Hulls

Because the supremum of a family of lower semicontinuous functions is lower semicontinuous, if is an arbitrary extended-real valued function on a topological space, the supremum of the set of lower semicontinuous functions majorized by is lower semicontinuous. This greatest lower semicontinuous function majorized by is the lower semicontinuous hull of. The hull is defined pointwise by the relation
The hull has the property that its epigraph is the closure of the epigraph of.
The lower semicontinuous hull plays a role in convex analysis. Given a convex function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is convex, and is known as the closure of the original convex function.
Some operations in convex analysis, such as the Legendre transform automatically produce closed convex functions. The Legendre transform applied twice to a convex function gives the closure of the original function, rather than the original function. Thus the lower semicontinuous hull is a way of regularizing convex functions, by modifying it at boundary points of its effective domain.
In categorical terms, the lower semicontinuous hull of a function is the Kan extension of along the inclusion of the poset of open neighborhoods into the topological space. Explicitly, the value of the hull at a point is given by the colimit:
which coincides with, the left Kan extension under the inclusion functor. In this formulation, the process of taking the semicontinuous envelope is a special case of the Kan extension machinery in enriched category theory. The upper semicontinuous hull is a right Kan extension.
Other types of hulls are often considered in applications. For example, the infimum of the set of continuous affine functions that majorize a given function on a convex subset of a topological vector space is upper semicontinuous. This fact is used in the proof of the Choquet theorem. Similar ideas applied to subharmonic functions are used in the Perron method for solving the Dirichlet problem for the Laplace operator in a domain. The key condition for the class of subharmonic solutions is upper semicontinuity, particularly near the boundary where the boundary conditions are applied.