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For | :''For the notion of upper or lower semicontinuous [[multivalued function]] see: [[Hemicontinuity]]'' | ||
In [[mathematical analysis]], '''semi-continuity''' (or '''semicontinuity''') is a property of [[extended real number|extended real]]-valued [[function (mathematics)|function]]s that is weaker than [[continuous function|continuity]]. An extended real-valued function ''f'' is '''upper''' (respectively, '''lower''') '''semi-continuous''' at a point ''x''<sub>0</sub> if, roughly speaking, the function values for arguments near ''x''<sub>0</sub> are either close to ''f''(''x''<sub>0</sub>) or less than (respectively, greater than) ''f''(''x''<sub>0</sub>). | |||
== Examples == | |||
[[Image:Upper semi.svg|thumb|right|An upper semi-continuous function. The solid blue dot indicates ''f''(''x''<sub>0</sub>).]] | |||
Consider the function ''f'', [[piecewise]] defined by ''f''(''x'') = –1 for ''x'' < 0 and ''f''(''x'') = 1 for ''x'' ≥ 0. This function is upper semi-continuous at ''x''<sub>0</sub> = 0, but not lower semi-continuous. | |||
[[Image:Lower semi.svg|thumb|right|A lower semi-continuous function. The solid blue dot indicates ''f''(''x''<sub>0</sub>).]] | |||
The [[indicator function]] of an [[open set]] is lower semi-continuous, whereas the indicator function of a [[closed set]] is upper semi-continuous. The [[floor function]] <math>f(x)=\lfloor x \rfloor</math>, which returns the greatest integer less than or equal to a given real number ''x'', is everywhere upper semi-continuous. Similarly, the [[ceiling function]] <math>f(x)=\lceil x \rceil</math> is lower semi-continuous. | |||
A function may be upper or lower semi-continuous without being either [[Continuous_function#Directional_and_semi-continuity|left or right continuous]]. For example, the function | |||
:<math>f(x) = \begin{cases} | |||
1 & \mbox{if } x < 1,\\ | |||
2 & \mbox{if } x = 1,\\ | |||
1/2 & \mbox{if } x > 1, | |||
\end{cases} </math> | |||
is upper semi-continuous at ''x'' = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function | |||
:<math> f(x) = \begin{cases} | |||
\sin(1/x) & \mbox{if } x \neq 0,\\ | |||
1 & \mbox{if } x = 0, | |||
\end{cases}</math> | |||
is upper semi-continuous at ''x'' = 0 while the function limits from the left or right at zero do not even exist. | |||
If <math>X=\mathbb R^n</math> is an Euclidean space (or more generally, a metric space) and <math>\Gamma=C([0,1],X)</math> is the space of [[curve]]s in <math>X</math> ( with the [[supremum norm|supremum distance]] <math>d_\Gamma(\alpha,\beta)=\sup_t\ d_X(\alpha(t),\beta(t))</math>, then the length functional <math>L:\Gamma\to[0,+\infty]</math>, which assigns to each curve <math>\alpha</math> its [[Curve#Length_of_curves|length]] <math>L(\alpha)</math>, is lower semicontinuous. | |||
Let <math>(X,\mu)</math> be a measure space and let <math>L^+(X,\mu)</math> denote the set of positive measurable functions endowed with the | |||
topology of [[convergence in measure]] with respect to <math>\mu</math>. Then the integral, seen as an operator from <math>L^+(X,\mu)</math> to | |||
<math>[-\infty,+\infty]</math> is lower semi-continuous. This is just [[Fatou's lemma]]. | |||
== Formal definition == | |||
Suppose ''X'' is a [[topological space]], ''x''<sub>0</sub> is a point in ''X'' and ''f'' : ''X'' → '''R''' ∪ {–∞,+∞} is an extended real-valued function. We say that ''f'' is '''upper semi-continuous''' at ''x''<sub>0</sub> if for every ε > 0 there exists a [[neighborhood (topology)|neighborhood]] ''U'' of ''x''<sub>0</sub> such that ''f''(''x'') ≤ ''f''(''x''<sub>0</sub>) + ε for all ''x'' in ''U'' when ''f''(''x''<sub>0</sub>) > -∞, and ''f''(''x'') tends to -∞ as ''x'' tends towards ''x''<sub>0</sub> when ''f''(''x''<sub>0</sub>) = -∞. For the particular case of a metric space, this can be expressed as | |||
:<math>\limsup_{x\to x_{0}} f(x)\le f(x_0)</math> | |||
where lim sup is the [[limit superior]] (of the function ''f'' at point ''x''<sub>0</sub>). (For non-metric spaces, an equivalent definition using [[net (mathematics)|net]]s may be stated.) | |||
The function ''f'' is called upper semi-continuous if it is upper semi-continuous at every point of its [[domain (function)|domain]]. A function is upper semi-continuous if and only if {''x'' ∈ ''X'' : ''f''(''x'') < α} is an [[open set]] for every α ∈ '''R'''. | |||
We say that ''f'' is '''lower semi-continuous''' at ''x''<sub>0</sub> if for every ε > 0 there exists a [[neighborhood (topology)|neighborhood]] ''U'' of ''x''<sub>0</sub> such that ''f''(''x'') ≥ ''f''(''x''<sub>0</sub>) – ε for all ''x'' in ''U'' when ''f''(''x''<sub>0</sub>) < +∞, and ''f''(''x'') tends to +∞ as ''x'' tends towards ''x''<sub>0</sub> when ''f''(''x''<sub>0</sub>) = +∞. Equivalently, this can be expressed as | |||
:<math>\liminf_{x\to x_0} f(x)\ge f(x_0)</math> | |||
where lim inf is the [[limit inferior]] (of the function ''f'' at point ''x''<sub>0</sub>). | |||
The function ''f'' is called lower semi-continuous if it is lower semi-continuous at every point of its domain. A function is lower semi-continuous if and only if {''x'' ∈ ''X'' : ''f''(''x'') > α} is an [[open set]] for every α ∈ '''R'''; alternatively, a function is lower semi-continuous if and only if all of its lower [[level set|levelset]]s {''x'' ∈ ''X'' : ''f''(''x'') ≤ α} are [[closed set|closed]]<!-- a formulation useful in [[optimization theory|minimization theory]], for [[Weierstrauss's theorem]] on the existence of a minimum for an inf-compact function, or for the definition of a quasi-convex function by the convexity of lower level sets, etc. -->. Lower level sets are also called ''[[level set|sublevel sets]]'' or ''trenches''.<ref>{{Cite news|last=Kiwiel|first=Krzysztof C.|title=Convergence and efficiency of subgradient methods for quasiconvex minimization|journal=Mathematical Programming (Series A)|publisher=Springer|location=Berlin, Heidelberg|issn=0025-5610|pages=1–25|volume=90|issue=1|doi=10.1007/PL00011414|year=2001|mr=1819784}}</ref> | |||
== Properties == | |||
A function is [[continuous function|continuous]] at ''x''<sub>0</sub> if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity. | |||
If ''f'' and ''g'' are two real-valued functions which are both upper semi-continuous at ''x''<sub>0</sub>, then so is ''f'' + ''g''. If both functions are non-negative, then the product function ''fg'' will also be upper semi-continuous at ''x''<sub>0</sub>. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function. | |||
If ''C'' is a [[compact space]] (for instance a [[closed set|closed]], [[bounded set|bounded]] [[interval (mathematics)|interval]] [''a'', ''b'']) and ''f'' : ''C'' → [–∞,∞) is upper semi-continuous, then ''f'' has a maximum on ''C''. The analogous statement for (–∞,∞]-valued lower semi-continuous functions and minima is also true. (See the article on the [[extreme value theorem]] for a proof.) | |||
Suppose ''f''<sub>''i''</sub> : ''X'' → [–∞,∞] is a lower semi-continuous function for every index ''i'' in a nonempty set ''I'', and define ''f'' as pointwise [[supremum]], i.e., | |||
:<math>f(x)=\sup_{i\in I}f_i(x),\qquad x\in X.</math> | |||
Then ''f'' is lower semi-continuous. Even if all the ''f''<sub>''i''</sub> are continuous, ''f'' need not be continuous: indeed every lower semi-continuous function on a [[uniform space]] (e.g. a [[metric space]]) arises as the supremum of a sequence of continuous functions. | |||
The [[indicator function]] of any open set is lower semicontinuous. The indicator function of a closed set is upper semicontinuous. However, in convex analysis, the term "indicator function" often refers to the [[Characteristic function (convex analysis)|characteristic function]], and the characteristic function of any ''closed'' set is lower semicontinuous, and the characteristic function of any ''open'' set is upper semicontinuous. | |||
A function ''f'' : '''R'''<sup>n</sup>→'''R''' is lower semicontinuous if and only if its [[epigraph (mathematics)|epigraph]] (the set of points lying on or above its [[graph of a function|graph]]) is [[Closed set|closed]]. | |||
A function ''f'' : ''X''→'''R''', for some topological space ''X'', is lower semicontinuous if and only if it is continuous with respect to the [[Scott topology]] on '''R'''. | |||
==References== | |||
{{Reflist}} | |||
*{{cite book | |||
| last = Bourbaki | |||
| first = Nicolas | |||
| title = Elements of Mathematics: General Topology, 1–4 | |||
| publisher = Springer | |||
| year = 1998 | |||
| pages = | |||
| isbn = 0-201-00636-7 | |||
}} | |||
*{{cite book | |||
| last = Bourbaki | |||
| first = Nicolas | |||
| title = Elements of Mathematics: General Topology, 5–10 | |||
| publisher = Springer | |||
| year = 1998 | |||
| pages = | |||
| isbn = 3-540-64563-2 | |||
}} | |||
*{{cite book | |||
| last = Gelbaum | |||
| first = Bernard R. | |||
| coauthors = Olmsted, John M.H. | |||
| title = Counterexamples in analysis | |||
| publisher = Dover Publications | |||
| year = 2003 | |||
| pages = | |||
| isbn = 0-486-42875-3 | |||
}} | |||
*{{cite book | |||
| last = Hyers | |||
| first = Donald H. | |||
| coauthors = Isac, George; Rassias, Themistocles M. | |||
| title = Topics in nonlinear analysis & applications | |||
| publisher = World Scientific | |||
| year = 1997 | |||
| pages = | |||
| isbn = 981-02-2534-2 | |||
}} | |||
==See also== | |||
* [[left-continuous|Directional continuity]] | |||
* [[Hemicontinuity|Semicontinuous multivalued function]] | |||
{{DEFAULTSORT:Semi-Continuity}} | |||
[[Category:Mathematical analysis]] | |||
[[Category:Variational analysis]] | |||
[[Category:Mathematical optimization]] |
Revision as of 15:42, 19 August 2013
- For the notion of upper or lower semicontinuous multivalued function see: Hemicontinuity
In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semi-continuous at a point x0 if, roughly speaking, the function values for arguments near x0 are either close to f(x0) or less than (respectively, greater than) f(x0).
Examples
Consider the function f, piecewise defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x0 = 0, but not lower semi-continuous.
The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function , which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.
A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function
is upper semi-continuous at x = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function
is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist.
If is an Euclidean space (or more generally, a metric space) and is the space of curves in ( with the supremum distance , then the length functional , which assigns to each curve its length , is lower semicontinuous.
Let be a measure space and let denote 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 semi-continuous. This is just Fatou's lemma.
Formal definition
Suppose X is a topological space, x0 is a point in X and f : X → R ∪ {–∞,+∞} is an extended real-valued function. We say that f is upper semi-continuous at x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≤ f(x0) + ε for all x in U when f(x0) > -∞, and f(x) tends to -∞ as x tends towards x0 when f(x0) = -∞. For the particular case of a metric space, this can be expressed as
where lim sup is the limit superior (of the function f at point x0). (For non-metric spaces, an equivalent definition using nets may be stated.)
The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain. A function is upper semi-continuous if and only if {x ∈ X : f(x) < α} is an open set for every α ∈ R.
We say that f is lower semi-continuous at x0 if for every ε > 0 there exists a neighborhood U of x0 such that f(x) ≥ f(x0) – ε for all x in U when f(x0) < +∞, and f(x) tends to +∞ as x tends towards x0 when f(x0) = +∞. Equivalently, this can be expressed as
where lim inf is the limit inferior (of the function f at point x0).
The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain. A function is lower semi-continuous if and only if {x ∈ X : f(x) > α} is an open set for every α ∈ R; alternatively, a function is lower semi-continuous if and only if all of its lower levelsets {x ∈ X : f(x) ≤ α} are closed. Lower level sets are also called sublevel sets or trenches.[1]
Properties
A function is continuous at x0 if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity.
If f and g are two real-valued functions which are both upper semi-continuous at x0, then so is f + g. If both functions are non-negative, then the product function fg will also be upper semi-continuous at x0. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.
If C is a compact space (for instance a closed, bounded interval [a, b]) and f : C → [–∞,∞) is upper semi-continuous, then f has a maximum on C. The analogous statement for (–∞,∞]-valued lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)
Suppose fi : X → [–∞,∞] is a lower semi-continuous function for every index i in a nonempty set I, and define f as pointwise supremum, i.e.,
Then f is lower semi-continuous. Even if all the fi are continuous, f need not be continuous: indeed every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions.
The indicator function of any open set is lower semicontinuous. The indicator function of a closed set is upper semicontinuous. However, in convex analysis, the term "indicator function" often refers to the characteristic function, and the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.
A function f : Rn→R is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.
A function f : X→R, for some topological space X, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on R.
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534