|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| :''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 ==
| | Would you answer a quick question? When you hear the words top skin care lines, do you think of [http://okgarage.org/forum/member.php?u=1684-WMcclary Clinique Skin Care] big name advertising or [http://search.Huffingtonpost.com/search?q=elite+companies&s_it=header_form_v1 elite companies] that spend their money on quality ingredients?<br><br>You may also have heard about the Baby Quasar Photo Rejuvenation Light Therapy device, which emits red light and reduce fine lines and wrinkles. It's the first product by Quasar which Baby Blue was dedicated to compliment. You can get each device separately according to your skin care issue, or both together with the Baby Quasar Power Pack. This could be a solution to consider for a complete skin rejuvenation.<br><br>A shortage of fluid circulation can cause dark circles around the eyes. Also by just improving circulation can get rid of eye puffiness. Eye wrinkles can be greatly reduced by using the correct eye cream ingredients.<br><br>Blemishes:This might sound odd, but using Visine (yes eye drops) will actually take away redness in your [http://www.Dict.cc/englisch-deutsch/pimples.html pimples] and acne. It will not clear it up, but it will lessen the appearance of it and make it easier to conceal.<br><br>If you workout, then it aids the heart and presents more oxygen to the skin hence helping it to retain its flexibility and as an added bonus will help decrease facial lines as well.<br><br>Today there are many types of wrinkle creams available. A few of these work well enough that people call them mini fountains of youth. Some will cause acne because they clog the pores. How can you tell which ones really work? Of course you don't want your skin issues to worsen right? This article will walk you through looking at all of the different wrinkle creams on the market and make objective opinions on each one. Keep reading if you really want the best creams available.<br><br>There you have it, the ultimate in dry skin relief. Try it for one week, then refer it to a friend...and throw away all the lotions and creams that not only don't work, but also contribute to the problem of dry skin with irritating chemicals and alcohols. |
| [[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]]
| |
Would you answer a quick question? When you hear the words top skin care lines, do you think of Clinique Skin Care big name advertising or elite companies that spend their money on quality ingredients?
You may also have heard about the Baby Quasar Photo Rejuvenation Light Therapy device, which emits red light and reduce fine lines and wrinkles. It's the first product by Quasar which Baby Blue was dedicated to compliment. You can get each device separately according to your skin care issue, or both together with the Baby Quasar Power Pack. This could be a solution to consider for a complete skin rejuvenation.
A shortage of fluid circulation can cause dark circles around the eyes. Also by just improving circulation can get rid of eye puffiness. Eye wrinkles can be greatly reduced by using the correct eye cream ingredients.
Blemishes:This might sound odd, but using Visine (yes eye drops) will actually take away redness in your pimples and acne. It will not clear it up, but it will lessen the appearance of it and make it easier to conceal.
If you workout, then it aids the heart and presents more oxygen to the skin hence helping it to retain its flexibility and as an added bonus will help decrease facial lines as well.
Today there are many types of wrinkle creams available. A few of these work well enough that people call them mini fountains of youth. Some will cause acne because they clog the pores. How can you tell which ones really work? Of course you don't want your skin issues to worsen right? This article will walk you through looking at all of the different wrinkle creams on the market and make objective opinions on each one. Keep reading if you really want the best creams available.
There you have it, the ultimate in dry skin relief. Try it for one week, then refer it to a friend...and throw away all the lotions and creams that not only don't work, but also contribute to the problem of dry skin with irritating chemicals and alcohols.