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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 which at least one input is associated with multiple (two or more) outputs. 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. ==Examples== *Every real number greater than zero has two real square roots. The square roots of 4 are in the set . The square root of 0 is 0. *Each complex number except zero has two square roots, three cube roots, and in general ''n'' nth roots. The nth root of 0 is 0. *The complex logarithm function is multiple-valued. The values assumed by for real numbers and are :As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan ''x'' to -π/2 < ''x'' < π/2 – a domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes -π/2 < ''y'' < π/2. These values from a restricted domain are called ''principal values''. * 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 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 ''n''th 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multivalued function」の詳細全文を読む スポンサード リンク
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