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{{About|multivalued functions as they are considered in mathematical analysis|set-valued functions as considered in variational analysis|set-valued function}}{{distinguish|Multivariate function}}
In [[mathematics]], a '''multivalued function''' is a [[set-valued function]] with additional properties depending on context. The terms '''multifunction''' and '''many-valued function''' are sometimes also used.
A ''multivalued function'' of sets ''f : X → Y'' is a subset
:<math> \Gamma_f\ \subseteq \ X\times Y.</math>
Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γ<sub>f</sub>''. Ordinary functions are multivalued functions by taking their [[Graph of a function|graph]]. They are called '''single-valued functions''' to distinguish them.
== Motivation ==▼
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.▼
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>.▼
==Inverses of functions==
If ''f : X → Y'' is an ordinary function, then its inverse the multivalued function
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Given any [[Holomorphic function|holomorphic]] function on an open subset of the [[Complex plane|complex plane]] '''C''', its [[analytic continuation]] is always a multivalued function.
==Concrete examples==
▲== Motivation ==
▲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.
▲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]] 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>.
*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.
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