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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
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 ...
-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.
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 points ...
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
of the function
f at point
x_0.
A function
f:X\to\overline is called lower semicontinuous if it satisfies any of the following equivalent conditions:
:(1) The function is lower semicontinuous at every point of its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
.
:(2) All sets
f^((y,\infty ])=\ with
y\in\R are open (topology), open in
X, where
(y,\infty ]=\.
:(3) All sublevel sets
\ with
y\in\R are closed (topology), closed in
X.
:(4) The
epigraph (mathematics), epigraph \ is closed in
X\times\R.
:(5) The function is continuous when the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
\overline is given the
right order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, th ...
. This is just a restatement of condition (2) since the right order topology is generated by all the intervals
(y,\infty ] .
Examples
Consider the function
f, piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
defined by:
f(x) = \begin
-1 & \mbox x < 0,\\
1 & \mbox x \geq 0
\end
This function is upper semicontinuous at
x_0 = 0, but not lower semicontinuous.
The
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
f(x) = \lfloor x \rfloor, which returns the greatest integer less than or equal to a given real number
x, is everywhere upper semicontinuous. Similarly, the
ceiling function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
f(x) = \lceil x \rceil 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
f(x) = \begin
\sin(1/x) & \mbox x \neq 0,\\
1 & \mbox x = 0,
\end
is upper semicontinuous at
x = 0 while the function limits from the left or right at zero do not even exist.
If
X = \R^n is a Euclidean space (or more generally, a metric space) and
\Gamma = C( ,1 X) is the space of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s in
X (with the
supremum distance d_\Gamma(\alpha,\beta) = \sup\), then the length functional
L : \Gamma \to , +\infty which assigns to each curve
\alpha its
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
L(\alpha), 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
\sqrt 2.
Let
(X,\mu) be a measure space and let
L^+(X,\mu) denote the set of positive measurable functions endowed with the
topology of
convergence in measure
Convergence in measure is either of two distinct mathematical concepts both of which generalize
the concept of convergence in probability.
Definitions
Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X, \ ...
with respect to
\mu. Then by
Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
Fatou's lemm ...
the integral, seen as an operator from
L^+(X,\mu) to
\infty, +\infty/math> is lower semicontinuous.
Properties
Unless specified otherwise, all functions below are from 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 points ...
X to 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 \overline= \infty,\infty/math>. Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated from semicontinuity over the whole domain.
* A function f:X\to\overline is continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
if and only if it is both upper and lower semicontinuous.
* The indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of a set A\subset X (defined by \mathbf_A(x)=1 if x\in A and 0 if x\notin A) is upper semicontinuous if and only if A is a closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
. It is lower semicontinuous if and only if A is an open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
.[
In the context of ]convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of s ...
, the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of a set A is defined differently, as \chi_(x)=0 if x\in A and \chi_A(x) = \infty if x\notin A. With that definition, the characteristic function of any is lower semicontinuous, and the characteristic function of any is upper semicontinuous.
* The sum f+g of two lower semicontinuous functions is lower semicontinuous (provided the sum is well-defined, i.e., f(x)+g(x) is not the indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this su ...
-\infty+\infty). The same holds for upper semicontinuous functions.
* If both functions are non-negative, the product function f g of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
* A function f:X\to\overline is lower semicontinuous if and only if -f is upper semicontinuous.
* The composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
f \circ g of upper semicontinuous functions is not necessarily upper semicontinuous, but if f is also non-decreasing, then f \circ g is upper semicontinuous.
* The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from X to \overline (or to \R) forms a lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
. The same holds for upper semicontinuous functions.
* The (pointwise) supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of an arbitrary family (f_i)_ of lower semicontinuous functions f_i:X\to\overline (defined by f(x)=\sup\) is lower semicontinuous.
:In particular, the limit of a monotone increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
sequence f_1\le f_2\le f_3\le\cdots of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions f_n(x)=1-(1-x)^n defined for x\in ,1/math> for n=1,2,....
:Likewise, the infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
sequence of continuous functions is upper semicontinuous.
* (Theorem of Baire)[ The result was proved by René Baire in 1904 for real-valued function defined on \R. It was extended to metric spaces by Hans Hahn in 1917, and ]Hing Tong
Hing Tong (16 February 1922 – 4 March 2007) was an American mathematician. He is well known for providing the original proof of the Katetov–Tong insertion theorem.
Life
Hing Tong was born in Canton, China. He received his bachelor's degree ...
showed in 1952 that the most general class of spaces where the theorem holds is the class of perfectly normal space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
s. (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.) Assume X is a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. Every lower semicontinuous function f:X\to\overline is the limit of a monotone increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
sequence of extended real-valued continuous functions on X; if f does not take the value -\infty, the continuous functions can be taken to be real-valued.
:And every upper semicontinuous function f:X\to\overline is the limit of a monotone decreasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
sequence of extended real-valued continuous functions on X; if f does not take the value \infty, the continuous functions can be taken to be real-valued.
* If C is a compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
(for instance a closed bounded interval , b/math>) and f:C\to\overline is upper semicontinuous, then f has a maximum on C. If f is lower semicontinuous on C, it has a minimum on C.
:(''Proof for the upper semicontinuous case'': By condition (5) in the definition, f is continuous when \overline is given the left order topology. So its image f(C) is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> suc ...
.)
* Any upper semicontinuous function f : X \to \mathbb on an arbitrary topological space X is locally constant on some dense open subset of X.
See also
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Notes
References
Bibliography
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{{DEFAULTSORT:Semi-Continuity
Theory of continuous functions
Mathematical analysis
Variational analysis