Multivalued function: Difference between revisions

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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]], '''multivalued function''', also called '''multifunction''' and '''many-valued function''', is a [[set-valued function]] with continuity properties that allow considering it locally as an ordinary function.

Multivalued functions arise commonly in applications of [[implicit function theorem]], since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the [[inverse function]] of a [[differentiable function]] is a multivalued function. For example, the [[complex logarithm]] is a multivalued function, as the inverse of the exponential function. It cannot be considered as an ordinary function, since, when one followfollows one value of the logarithm along a circle centered at {{math|0}}, one gets another value than the starting one after a complete turn. This phenomenon is called [[monodromy]].

Another common way for defining a multivalued function is [[analytic continuation]], which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value.