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Multivalued function
Function  

x ↦ f(x)  
By domain and codomain  


Classes/properties  
Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective  
Constructions  
Restriction · Composition · λ · Inverse  
Generalizations  
Partial · Multivalued · Implicit  
In mathematics, a multivalued function (short form: multifunction; other names: manyvalued function, setvalued function, setvalued map, pointtoset map, multivalued map, multimap, correspondence, carrier) is a lefttotal relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.^{[citation needed]}
In the strict sense, a "welldefined" function associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a misnomer because functions are singlevalued. 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.
Contents
Examples
 Every real number greater than zero has two real square roots. The square roots of 4 are in the set {+2,−2}. The square root of 0 is 0.
 Each complex number except zero has two square roots, three cube roots, and in general n nth roots. The nth root of 0 is 0.
 The complex logarithm function is multiplevalued. 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 integers <math>n</math>.
 Inverse trigonometric functions are multiplevalued because trigonometric functions are periodic. We have
 <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>
 As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a singlevalued 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 values.
 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 noninjective 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 points. For example, for the nth 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 singlevalued 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.
Setvalued analysis
Setvalued analysis is the study of sets in the spirit of mathematical analysis and general topology.
Instead of considering collections of only points, setvalued 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 setvalued 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 subdifferentials to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.
There exist setvalued extensions of the following concepts from pointvalued analysis: continuity, differentiation, integration, implicit function theorem, contraction mappings, measure theory, fixedpoint theorems, optimization, and topological degree theory.
Equations are generalized to inclusions.
Types of multivalued functions
One can differentiate many continuity concepts, primarily closed graph property and 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 upper and lower semicontinuous realvalued 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 functions, 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
Multifunctions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed point theorem for multifunctions has been applied to prove existence of Nash equilibria (note: in the context of game theory, a multivalued function is usually referred to as a 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.
Nevertheless, lower hemicontinuous multifunctions usually possess continuous selections as stated in the Michael selection theorem, which provides another characterisation of paracompact spaces.^{[1]}^{[2]} Other selection theorems, like BressanColombo directional continuous selection, Kuratowski—RyllNardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in optimal control and the theory of differential inclusions.
In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystals and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of gauge field structures in many branches of physics.^{[citation needed]}
Contrast with
References
 ^ Ernest Michael (Mar 1956). "Continuous Selections. I" (PDF). The Annals of Mathematics, 2nd Ser. 63 (2): 361—382.
 ^ Dušan Repovš; P.V. Semenov (2008). "Ernest Michael and theory of continuous selections". Topology Appl. 155 (8): 755—763. arXiv:0803.4473v1.
 C. D. Aliprantis and K. C. Border, Infinite dimensional analysis. Hitchhiker's guide, SpringerVerlag Berlin Heidelberg, 2006
 J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, 2003
 J.P. Aubin and A. Cellina, Differential Inclusions, SetValued Maps And Viability Theory, Grundl. der Math. Wiss. 264, Springer  Verlag, Berlin, 1984
 J.P. Aubin and H. Frankowska, SetValued Analysis, Birkhäuser, Basel, 1990
 K. Deimling, Multivalued Differential Equations, Walter de Gruyter, 1992
 A. Geletu, Introduction to Topological Spaces and SetValued Maps (Lecture notes), Ilmenau University of Technology, 2006
 H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online)
 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: Vol. I and Vol. II)
 D. Repovš and P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998
 E. U. Tarafdar and M. S. R. Chowdhury, Topological methods for setvalued nonlinear analysis, World Scientific, Singapore, 2008