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| [[File:Multivalued function.svg|frame|right|This diagram does not represent a "true" [[function (mathematics)|function]], because the element 3 in ''X'' is associated with two elements, ''b'' and ''c'', in ''Y''.]]
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| In [[mathematics]], a '''multivalued function''' (shortly: '''multifunction''', other names: '''many-valued function''', '''set-valued function''', '''set-valued map''', '''multi-valued map''', '''multimap''', '''correspondence''', '''carrier''') is a [[binary relation|left-total relation]]; that is, every [[Input/output|input]] is associated with at least one [[output]].
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| Strictly speaking, a "well-defined" [[function (mathematics)|function]] associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a [[misnomer]] because functions are single-valued. Multivalued functions often arise from functions which are not [[injective]]. Such functions do not have an [[inverse function]], but they do have an [[inverse relation]]. The multivalued function corresponds to this inverse relation.
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| ==Examples==
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| *Every [[real number|real]] number greater than zero has two [[square root]]s. The square roots of 4 are in the set {+2,−2}. The square root of 0 is 0.
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| *Each [[complex number]] except zero has two square roots, three [[cube root]]s, and in general ''n'' [[nth roots]]. The [[nth root]] of 0 is 0.
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| *The [[complex logarithm]] function is multiple-valued. The values assumed by log(1) are <math>2 \pi n i</math> for all [[integer]]s <math>n</math>.
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| *[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have
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| ::<math>
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| \tan\left({\textstyle\frac{\pi}{4}}\right) = \tan\left({\textstyle\frac{5\pi}{4}}\right)
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| = \tan\left({\textstyle\frac{-3\pi}{4}}\right) = \tan\left({\textstyle\frac{(2n+1)\pi}{4}}\right) = \cdots = 1.
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| </math> | |
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| :Consequently arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan ''x'' to -π/2 < ''x'' < π/2 – a domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes -π/2 < ''y'' < π/2. These values from a restricted domain are called ''[[principal value]]s''.
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| * The [[indefinite integral]] can be considered as a multivalued function. The indefinite integral of a function is the set of functions whose derivative is that function. The [[constant of integration]] follows from the fact that the derivative of a constant function is 0.
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| These are all examples of multivalued functions which come about from non-[[injective]] functions. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a [[partial inverse]] of the original function.
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| Multivalued functions of a complex variable have [[branch point]]s. For example the ''n''th root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units ''i'' and −''i'' are branch points. Using the branch points these functions may be redefined to be single valued functions, by restricting the range. A suitable interval may be found through use of a [[branch cut]], a kind of curve which connects pairs of branch points, thus reducing the multilayered [[Riemann surface]] of the function to a single layer. As in the case with real functions the restricted range may be called ''principal branch'' of the function.
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| ==Set-valued analysis==
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| '''Set-valued analysis''' is the study of sets in the spirit of [[mathematical analysis]] and [[general topology]].
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| Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied.
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| Much of set-valued analysis arose through the study of [[mathematical economics]] and [[optimal control]], partly as a generalization of [[convex analysis]]; the term "[[variational analysis]]" is used by authors such as [[R. T. Rockafellar]] and [[Roger Wets]], [[Jon Borwein]] and [[Adrian Lewis]], and [[Boris Mordukhovich]]. In optimization theory, the convergence of approximating [[subdifferential]]s to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.
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| There exist set-valued extensions of the following concepts from point-valued analysis: [[continuous (mathematics)|continuity]], [[differentiation (mathematics)|differentiation]], [[integral|integration]], [[implicit function theorem]], [[contraction mapping]]s, [[measure theory]], [[fixed-point theorem]]s, [[Optimization (mathematics)|optimization]], and [[topological degree theory]].
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| [[Equation]]s are generalized to [[Inclusion (set theory)|inclusions]].
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| ==Types of multivalued functions==
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| One can differentiate many continuity concepts, primarily closed graph property and [[Hemicontinuity|upper and lower hemicontinuity]]. (One should be warned that often the terms upper and lower semicontinuous are used instead of upper and lower hemicontinuous reserved for the case of weak topology in domain; yet we arrive at the collision with the reserved names for [[Semicontinuity|upper and lower semicontinuous]] real-valued function). There exist also various definitions for measurability of multifunction.
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| ==History==
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| The practice of allowing ''function'' in mathematics to mean also ''multivalued function'' dropped out of usage at some point in the first half of the twentieth century. Some evolution can be seen in different editions of ''[[A Course of Pure Mathematics]]'' by [[G. H. Hardy]], for example. It probably persisted longest in the theory of [[special function]]s, for its occasional convenience.
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| The theory of multivalued functions was fairly systematically developed for the first time
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| in [[Claude Berge]]'s ''Topological spaces'' (1963).
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| ==Applications==
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| Multifunctions arise in [[Optimal control|optimal control theory]], especially [[differential inclusion]]s and related subjects as [[game theory]], where the [[Kakutani fixed point theorem]] for multifunctions has been applied to prove existence of [[Nash equilibrium|Nash equilibria]] (note: in the context of game theory, a multivalued function is usually referred to as a [[correspondence (mathematics)|correspondence]].) This amongst many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.
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| Nevertheless, lower hemicontinuous multifunctions usually possess continuous selections as stated in the [[Michael selection theorem]] which provides another characterisation of [[paracompact]] spaces (see: E. Michael, Continuous selections I" Ann. of Math. (2) 63 (1956), and D. Repovs, P.V. Semenov, Ernest Michael and theory of continuous selections" arXiv:0803.4473v1). Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski—Ryll-Nardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in [[optimal control]] and the theory of [[differential inclusion]]s.
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| In physics, multivalued functions play an increasingly
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| important role. They form the mathematical basis for [[Paul Dirac|Dirac]]'s [[magnetic monopole]]s, for the theory
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| of [[Crystallographic defect|defect]]s in crystal and the resulting [[Plasticity (physics)|plasticity]] of materials,
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| for [[vortex|vortices]] in [[superfluid]]s and [[superconductor]]s, and for [[phase transition]]s in these systems, for instance [[melting]] and [[quark confinement]].
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| They are the origin of [[gauge field]] structures in many branches of physics.{{Citation needed|reason=reliable source needed for the paragraph|date=July 2013}}
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| ==Contrast with==
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| {{expand list|date=March 2013}}
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| * [[Bijection]]
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| * [[Injection]]
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| ==References==
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| * Jean-Pierre Aubin, Arrigo Cellina ''Differential Inclusions, Set-Valued Maps And Viability Theory'', Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984
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| * J.-P. Aubin and H. Frankowska ''Set-Valued Analysis'', Birkhäuser, Basel, 1990
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| * Klaus Deimling ''Multivalued Differential Equations'', Walter de Gruyter, 1992
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| * [[Hagen Kleinert|Kleinert, Hagen]], ''Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation'', [http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 online])
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| * [[Hagen Kleinert|Kleinert, Hagen]], ''Gauge Fields in Condensed Matter'', Vol. I, "SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456, [http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)]; Paperback ISBN 9971-5-0210-0 '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II])''
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| * Aliprantis, Kim C. Border ''Infinite dimensional analysis. Hitchhiker's guide'' Springer
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| * J. Andres, L. Górniewicz ''Topological Fixed Point Principles for Boundary Value Problems'', Kluwer Academic Publishers, 2003
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| *[http://books.google.co.uk/books?id=Cir88lF64xIC Topological methods for set-valued nonlinear analysis], Enayet U. Tarafdar, Mohammad Showkat Rahim Chowdhury, World Scientific, 2008, ISBN 978-981-270-467-2
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| * Abebe Geletu: "Introduction to Topological Spaces and Set-Valued Maps" (Lecture notes:[https://www.tu-ilmenau.de/fileadmin/media/simulation/Lehre/Vorlesungsskripte/Lecture_materials_Abebe/svm-topology.pdf svm-topology.pdf]
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| ==See also==
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| * [[Partial function]]
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| * [[Correspondence (mathematics)|correspondence]]
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| * [[Fat link]], a one-to-many [[hyperlink]]
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| * [[Interval finite element]]
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| * [[Hans Rådström]]
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| [[Category:Functions and mappings]]
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