Multivalued function: Difference between revisions

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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''.]]

In [[mathematics]], a '''multivalued function''' (shortlyshort 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]].

StrictlyIn speakingthe 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 from functions whichthat 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.

==Examples==

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:ConsequentlyAs 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''.

* 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 whichthat 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 whichthat 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==

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==Applications==

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 amongstamong 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 (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.

In physics, multivalued functions play an increasingly