Multivaluedness

Multivaluedness is the property of a relation or mapping that assigns more than one output to a given input. In mathematics, a multivalued function F from a set X to a set Y associates to each x in X a nonempty subset F(x) of Y, rather than a single element. This occurs naturally when inverting a relation or solving equations where multiple solutions exist.

Classic examples include the square root and logarithm in the complex plane. For any nonzero z, the equation w^2 = z has two solutions, so the square root function is multivalued. Likewise, the complex logarithm log z has infinitely many values differing by multiples of 2πi. In practice, multivaluedness is often represented by a relation or as a set-valued map F: X ⇒ Y.

To work with multivalued objects, mathematicians use single-valued selections when possible or frame the problem in terms of set-valued analysis. In some contexts, a multivalued function is viewed as a relation, or as a map into the power set of Y. In analysis, differential inclusions use set-valued right-hand sides to model systems with uncertainty or non-deterministic behavior.

In complex analysis and topology, multivaluedness is studied through branch points, branch cuts, and Riemann surfaces, which organize different values into a single analytic object on a covering space. Monodromy describes how values permute when encircling a singularity, illustrating the global structure behind local single-valuedness.

Multivaluedness thus reflects non-uniqueness in inversion or solving equations, and its proper handling is essential across mathematics and related fields.