Multi-valued function

In mathematics, a multivalued function (short form: multifunction; other names: many-valued function, set-valued function, set-valued map, point-to-set map, multi-valued map, multimap, correspondence, carrier) is a left-total relation (that is, every input is associated with at least one output).

In the strict sense, a well-defined 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 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.

These are all examples of multivalued functions that 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 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 single-valued 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.

Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology.

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