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| :''For the notion of upper or lower semicontinuous [[multivalued function]] see: [[Hemicontinuity]]''
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| 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>).
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| == Examples ==
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| [[Image:Upper semi.svg|thumb|right|An upper semi-continuous function. The solid blue dot indicates ''f''(''x''<sub>0</sub>).]]
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| 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.
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| [[Image:Lower semi.svg|thumb|right|A lower semi-continuous function. The solid blue dot indicates ''f''(''x''<sub>0</sub>).]]
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| 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.
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| 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
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| :<math>f(x) = \begin{cases}
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| 1 & \mbox{if } x < 1,\\
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| 2 & \mbox{if } x = 1,\\
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| 1/2 & \mbox{if } x > 1,
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| \end{cases} </math>
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| 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
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| :<math> f(x) = \begin{cases}
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| \sin(1/x) & \mbox{if } x \neq 0,\\
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| 1 & \mbox{if } x = 0,
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| \end{cases}</math>
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| is upper semi-continuous at ''x'' = 0 while the function limits from the left or right at zero do not even exist.
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| 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.
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| 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
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| 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
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| <math>[-\infty,+\infty]</math> is lower semi-continuous. This is just [[Fatou's lemma]].
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| == Formal definition ==
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| 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
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| :<math>\limsup_{x\to x_{0}} f(x)\le f(x_0)</math>
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| 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.)
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| 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'''.
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| 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
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| :<math>\liminf_{x\to x_0} f(x)\ge f(x_0)</math>
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| where lim inf is the [[limit inferior]] (of the function ''f'' at point ''x''<sub>0</sub>).
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| 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>
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| == Properties ==
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| 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.
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| 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.
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| 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.)
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| 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.,
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| :<math>f(x)=\sup_{i\in I}f_i(x),\qquad x\in X.</math>
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| 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.
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| 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.
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| 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]].
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| 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'''.
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| ==References==
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| {{Reflist}}
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| *{{cite book
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| | last = Bourbaki
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| | first = Nicolas
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| | title = Elements of Mathematics: General Topology, 1–4
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| | publisher = Springer
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| | year = 1998
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| | pages =
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| | isbn = 0-201-00636-7
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| }}
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| *{{cite book
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| | last = Bourbaki
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| | first = Nicolas
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| | title = Elements of Mathematics: General Topology, 5–10
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| | publisher = Springer
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| | year = 1998
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| | pages =
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| | isbn = 3-540-64563-2
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| }}
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| *{{cite book
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| | last = Gelbaum
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| | first = Bernard R.
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| | coauthors = Olmsted, John M.H.
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| | title = Counterexamples in analysis
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| | publisher = Dover Publications
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| | year = 2003
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| | pages =
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| | isbn = 0-486-42875-3
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| }}
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| *{{cite book
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| | last = Hyers
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| | first = Donald H.
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| | coauthors = Isac, George; Rassias, Themistocles M.
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| | title = Topics in nonlinear analysis & applications
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| | publisher = World Scientific
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| | year = 1997
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| | pages =
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| | isbn = 981-02-2534-2
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| }}
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| ==See also==
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| * [[left-continuous|Directional continuity]]
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| * [[Hemicontinuity|Semicontinuous multivalued function]]
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| {{DEFAULTSORT:Semi-Continuity}}
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| [[Category:Mathematical analysis]]
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| [[Category:Variational analysis]]
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| [[Category:Mathematical optimization]]
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The author is known as Betty Digennaro In Guam I Have been for some time Badge collecting is what love doing. Choosing has been my day-job for a while and Iam doing very good economically
Here is my page - Clinique Skin Care