Multivalued function: Difference between revisions

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{{Short description|Generalized mathematical function}}

{{Other uses of|one-to-many}}

{{More footnotes needed|date=January 2020}}

{{About|multivalued functions as they are considered in mathematical analysis|set-valued functions as considered in variational analysis|set-valued function}}{{distinguish|Multivariate function}}

[[File:Multivalued_function.svg|thumb|Multivalued function {1,2,3} → {a,b,c,d}.]]

[[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''.]]

In [[mathematics]], a '''multivalued function''',<ref>{{Cite web |title=Multivalued Function |url=https://archive.lib.msu.edu/crcmath/math/math/m/m450.htm |access-date=2024-10-25 |website=archive.lib.msu.edu}}</ref> '''multiple-valued function''',<ref>{{Cite web |title=Multiple Valued Functions {{!}} Complex Variables with Applications {{!}} Mathematics |url=https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-fall-1999/pages/study-materials/multiple-valued-functions/ |access-date=2024-10-25 |website=MIT OpenCourseWare |language=en}}</ref> '''many-valued function''',<ref>{{Cite journal |last1=Al-Rabadi |first1=Anas |last2=Zwick |first2=Martin |date=2004-01-01 |title=Modified Reconstructability Analysis for Many-Valued Functions and Relations |url=https://pdxscholar.library.pdx.edu/sysc_fac/30/ |journal=Kybernetes |volume=33 |issue=5/6 |pages=906–920 |doi=10.1108/03684920410533967}}</ref> or '''multifunction''',<ref>{{Cite journal |last1=Ledyaev |first1=Yuri |last2=Zhu |first2=Qiji |date=1999-09-01 |title=Implicit Multifunction Theorems |url=https://scholarworks.wmich.edu/math_pubs/22/ |journal=Set-Valued Analysis Volume |volume=7 |issue=3 |pages=209–238|doi=10.1023/A:1008775413250 |url-access=subscription }}</ref> is a function that has two or more values in its range for at least one point in its ___domain.<ref>{{cite web |title=Multivalued Function |url=https://mathworld.wolfram.com/MultivaluedFunction.html |website=Wolfram MathWorld |access-date=10 February 2024}}</ref> It is a [[set-valued function]] with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,<ref>{{Cite book |last=Repovš |first=Dušan |title=Continuous selections of multivalued mappings |date=1998 |publisher=Kluwer Academic |others=Pavel Vladimirovič. Semenov |isbn=0-7923-5277-7 |___location=Dordrecht |oclc=39739641}}</ref> but English Wikipedia currently does, having a separate article for each.

A ''multivalued function'' of sets ''f : X → Y'' is a subset

In [[mathematics]], a '''multivalued function''' (short form: '''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]].

:<math> \Gamma_f\ \subseteq \ X\times Y.</math>

Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γ_f''. If ''f'' is an ordinary function, it is a multivalued function by taking its [[Graph of a function|graph]]

:<math> \Gamma_f\ =\ \{(x,f(x))\ :\ x\in X\}.</math>

They are called '''single-valued functions''' to distinguish them.

== Motivation ==

In the strict sence, 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 as inverses of functions that 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.

{{Main|Global analytic function}}

The term multivalued function originated in complex analysis, from [[analytic continuation]]. It often occurs that one knows the value of a complex [[analytic function]] <math>f(z)</math> in some [[neighbourhood (mathematics)|neighbourhood]] of a point <math>z=a</math>. This is the case for functions defined by the [[implicit function theorem]] or by a [[Taylor series]] around <math>z=a</math>. In such a situation, one may extend the ___domain of the single-valued function <math>f(z)</math> along curves in the complex plane starting at <math>a</math>. In doing so, one finds that the value of the extended function at a point <math>z=b</math> depends on the chosen curve from <math>a</math> to <math>b</math>; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.

For example, let <math>f(z)=\sqrt{z}\,</math> be the usual [[square root]] function on positive real numbers. One may extend its ___domain to a neighbourhood of <math>z=1</math> in the complex plane, and then further along curves starting at <math>z=1</math>, so that the values along a given curve vary continuously from <math>\sqrt{1}=1</math>. Extending to negative real numbers, one gets two opposite values for the square root—for example {{math|±''i''}} for {{math|−1}}—depending on whether the ___domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for [[nth root|{{mvar|n}}th roots]], [[logarithm]]s, and [[inverse trigonometric function]]s.

==Examples==

To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the [[principal value]], producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path ([[monodromy]]). These problems are resolved in the theory of [[Riemann surface]]s: to consider a multivalued function <math>f(z)</math> as an ordinary function without discarding any values, one multiplies the ___domain into a many-layered [[Branched covering|covering space]], a [[manifold]] which is the Riemann surface associated to <math>f(z)</math>.

*Every [[real number|real]] number greater than zero has two [[square root]]s. The square roots of 4 are in the set {+2,&minus;2}. The square root of 0 is 0.

==Inverses of functions==

*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.

If ''f : X → Y'' is an ordinary function, then its inverse is the multivalued function

*The [[complex logarithm]] function is multiple-valued. The values assumed by <math>\log(a+bi)</math> for real numbers <math>a</math> and <math>b</math> are <math>\arg (a+bi) + 2 \pi n i</math> for all [[integer]]s <math>n</math>.

:<math> \Gamma_{f^{-1}}\ \subseteq \ Y\times X</math>

defined as ''Γ_f'', viewed as a subset of ''X'' × ''Y''. When ''f'' is a [[differentiable function]] between [[Manifold|manifolds]], the [[inverse function theorem]] gives conditions for this to be single-valued locally in ''X''.

For example, the [[complex logarithm]] ''log(z)'' is the multivalued inverse of the exponential function ''e^z'' : '''C''' → '''C'''^×, with graph

:<math> \Gamma_{\log(z)}\ =\ \{(z,w)\ :\ w=\log (z)\}\ \subseteq\ \mathbf{C}\times\mathbf{C}^\times.</math>

It is not single valued, given a single ''w'' with ''w = log(z)'', we have

:<math>\log(z)\ =\ w\ +\ 2\pi i \mathbf{Z}.</math>

Given any [[Holomorphic function|holomorphic]] function on an open subset of the [[complex plane]] '''C''', its [[analytic continuation]] is always a multivalued function.

==Concrete examples==

*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have

*Every [[real number]] greater than zero has two real [[square root]]s, so that square root may be considered a multivalued function. For example, we may write <math>\sqrt{4}=\pm 2=\{2,-2\}</math>; although zero has only one square root, <math>\sqrt{0} =\{0\}</math>. Note that <math>\sqrt{x}</math> usually denotes only the principal square root of <math>x</math>.

*Each nonzero [[complex number]] has two square roots, three [[cube root]]s, and in general ''n'' [[nth root|''n''th roots]]. The only ''n''th root of 0 is 0.

*The [[complex logarithm]] function is multiple-valued. The values assumed by <math>\log(a+bi)</math> for real numbers <math>a</math> and <math>b</math> are <math>\log{\sqrt{a^2 + b^2}} + i\arg (a+bi) + 2 \pi n i</math> for all [[integer]]s <math>n</math>.

*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have <math display="block">

\tan\left(\tfrac{\pi}{4}\right) = \tan\left(\tfrac{5\pi}{4}\right)

= \tan\left({\tfrac{-3\pi}{4}}\right) = \tan\left({\tfrac{(2n+1)\pi}{4}}\right) = \cdots = 1.

</math> As a consequence, arctan(1) is intuitively related to several values: {{pi}}/4, 5{{pi}}/4, −3{{pi}}/4, and so on. We can treat arctan as a single-valued function by restricting the ___domain of tan ''x'' to {{nowrap|−{{pi}}/2 < ''x'' < {{pi}}/2}} – a ___domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes {{nowrap|−{{pi}}/2 < ''y'' < {{pi}}/2}}. These values from a restricted ___domain are called ''[[principal value]]s''.

* The [[antiderivative]] can be considered as a multivalued function. The antiderivative 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.

*[[Inverse hyperbolic functions]] over the complex ___domain are multiple-valued because hyperbolic functions are periodic along the imaginary axis. Over the reals, they are single-valued, except for arcosh and arsech.

These are all examples of multivalued functions that come about from non-[[injective function]]s. 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.

::<math>

\tan\left({\textstyle\frac{\pi}{4}}\right) = \tan\left({\textstyle\frac{5\pi}{4}}\right)

= \tan\left({\textstyle\frac{-3\pi}{4}}\right) = \tan\left({\textstyle\frac{(2n+1)\pi}{4}}\right) = \cdots = 1.

</math>

== Branch points ==

:As a consequence, arctan(1) is intuitively related to several values: &pi;/4, 5&pi;/4, &minus;3&pi;/4, and so on. We can treat arctan as a single-valued function by restricting the ___domain of tan ''x'' to -&pi;/2 < ''x'' < &pi;/2 – a ___domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes -&pi;/2 < ''y'' < &pi;/2. These values from a restricted ___domain are called ''[[principal value]]s''.

{{Main articles|Branch point}}

Multivalued functions of a complex variable have [[branch point]]s. For example, for the ''n''th root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units ''i'' and &minus;''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 that 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 the ''principal branch'' of the function.

* 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.

These are all examples of multivalued functions that 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.

Multivalued functions of a complex variable have [[branch point]]s. For example, for the ''n''th root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units ''i'' and &minus;''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 that 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.

==Set-valued analysis==

'''Set-valued analysis''' is the study of sets in the spirit of [[mathematical analysis]] and [[general topology]].

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.

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.

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]].

[[Equation]]s are generalized to [[Inclusion (set theory)|inclusions]].

==Types of multivalued functions==

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.

==History==

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.

The theory of multivalued functions was fairly systematically developed for the first time

in [[Claude Berge]]'s ''Topological spaces'' (1963).

==Applications==

In physics, multivalued functions play an increasingly important role. They form the mathematical basis for [[Paul Dirac|Dirac]]'s [[magnetic monopole]]s, for the theory of [[Crystallographic defect|defect]]s in crystals and the resulting [[Plasticity (physics)|plasticity]] of materials, for [[vortex|vortices]] in [[superfluid]]s and [[superconductor]]s, and for [[phase transition]]s in these systems, for instance [[melting]] and [[quark confinement]]. 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}}

== See also ==

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 among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.

* [[Relation (mathematics)]]

* [[Function (mathematics)]]

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&mdash;Ryll-Nardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in [[optimal control]] and the theory of [[differential inclusion]]s.

* [[Binary relation]]

* [[Set-valued function]]

In physics, multivalued functions play an increasingly

important role. They form the mathematical basis for [[Paul Dirac|Dirac]]'s [[magnetic monopole]]s, for the theory

of [[Crystallographic defect|defect]]s in crystal and the resulting [[Plasticity (physics)|plasticity]] of materials,

for [[vortex|vortices]] in [[superfluid]]s and [[superconductor]]s, and for [[phase transition]]s in these systems, for instance [[melting]] and [[quark confinement]].

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}}

==Contrast with==

* [[Bijection]]

* [[Injection]]

* [[Surjection]]

==References==

==Further reading==

* Jean-Pierre Aubin, Arrigo Cellina ''Differential Inclusions, Set-Valued Maps And Viability Theory'', Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984

* [[Hagen Kleinert|H. Kleinert]], ''Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation'', [https://web.archive.org/web/20080315225354/http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 online])

* J.-P. Aubin and H. Frankowska ''Set-Valued Analysis'', Birkhäuser, Basel, 1990

* [[Hagen Kleinert|H. Kleinert]], ''Gauge Fields in Condensed Matter'', Vol. I: Superflow and Vortex Lines, 1–742, Vol. II: Stresses and Defects, 743–1456, World Scientific, Singapore, 1989 (also available online: [http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II])

* Klaus Deimling ''Multivalued Differential Equations'', Walter de Gruyter, 1992

* [[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])

* [[Hagen Kleinert|Kleinert, Hagen]], ''Gauge Fields in Condensed Matter'', Vol. I, "SUPERFLOW AND VORTEX LINES", pp.&nbsp;1–742, Vol. II, "STRESSES AND DEFECTS", pp.&nbsp;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])''

* Aliprantis, Kim C. Border ''Infinite dimensional analysis. Hitchhiker's guide'' Springer

* J. Andres, L. Górniewicz ''Topological Fixed Point Principles for Boundary Value Problems'', Kluwer Academic Publishers, 2003

*[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

* 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]

==See alsoReferences ==

{{Reflist}}

* [[Partial function]]

* [[Correspondence (mathematics)|correspondence]]

* [[Fat link]], a one-to-many [[hyperlink]]

* [[Interval finite element]]

* [[Hans Rådström]]

[[Category:Functions and mappings]]

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