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Concept
Multivalued function
Summary
In mathematics, a 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 the 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, and is single-valued only when the original function is monotonic. 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 follows one value of the logarithm along a circle centered at 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.
Multivalued functions arise also as solutions of differential equations, where the different values are parametrized by initial conditions.
The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function in some neighbourhood of a point . This is the case for functions defined by the implicit function theorem or by a Taylor series around . In such a situation, one may extend the domain of the single-valued function along curves in the complex plane starting at . In doing so, one finds that the value of the extended function at a point depends on the chosen curve from to ; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.
For example, let be the usual square root function on positive real numbers.
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