Lower semicontinuous
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In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right). 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 x_0 to f\left(x_0\right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to f\left(x_0\right) - c 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 X 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 poin ...
and f:X\to\overline is a function with values in the extended real numbers \overline=\R \cup \ = \infty,\infty/math>.


Upper semicontinuity

A function f:X\to\overline is called upper semicontinuous at a point x_0 \in X if for every real y > f\left(x_0\right) there exists a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
U of x_0 such that f(x) for all x\in U.Stromberg, p. 132, Exercise 4 Equivalently, f is upper semicontinuous at x_0 if and only if \limsup_ f(x) \leq f(x_0) where lim sup is the
limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
of the function f at the point x_0. A function f:X\to\overline is called upper semicontinuous if it satisfies any of the following equivalent conditions: :(1) The function is upper 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 * ...
. :(2) All sets f^(
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
in X, where [ -\infty ,y)=\. :(3) All superlevel sets \ with y\in\R are closed (topology), closed in X. :(4) The hypograph (mathematics), hypograph \ 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 th ...
\overline is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals neighborhood_ A_neighbourhood_(British_English,_Irish_English,_Australian_English_and_Canadian_English)_or_neighborhood_(American_English;__see_spelling_differences)_is_a_geographically_localised_community_within_a_larger_city,_town,_suburb_or_rural_area,__...
_U_of_x_0_such_that_f(x)>y_for_all_x\in_U. Equivalently,_f_is_lower_semicontinuous_at_x_0_if_and_only_if \liminf__f(x)_\ge_f(x_0) where_\liminf_is_the_limit_inferior_(topological_space).html" ;"title="-\infty,y).


Lower semicontinuity

A function f:X\to\overline is called lower semicontinuous at a point x_0\in X if for every real y < f\left(x_0\right) there exists a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
U of x_0 such that f(x)>y for all x\in U. Equivalently, f is lower semicontinuous at x_0 if and only if \liminf_ f(x) \ge f(x_0) where \liminf is the limit inferior (topological space)">limit inferior 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 * ...
. :(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 th ...
\overline is given the right order topology. 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. P ...
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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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 Inte ...
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 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 le ...
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 poin ...
X to the extended real numbers \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 g ...
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 ...
. 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 su ...
. 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 som ...
, 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 s ...
-\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 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. 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 ...
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 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 lo ...
of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing 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 setti ...
. Every lower semicontinuous function f:X\to\overline is the limit of a monotone increasing 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 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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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> s ...
.) * 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

* * * * * * * * * {{DEFAULTSORT:Semi-Continuity Theory of continuous functions Mathematical analysis Variational analysis