Multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a codomain associates each in to one or more values in ; it is thus a serial binary relation. Some authors allow a multivalued function to have no value for some inputs (in this case a multivalued function is simply a binary relation). However, in some contexts such as in complex analysis ("X" = "Y" = C), authors prefer to mimic function theory  as they extend concepts of the ordinary (single-valued) functions. In this context, an ordinary function is often called a single-valued function to avoid confusion. The term "multivalued function" originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function formula_1 in some neighbourhood of a point formula_2. This is the case for functions defined by the implicit function theorem or by a Taylor series around formula_2. In such a situation, one may extend the domain of the single-valued function formula_1 along curves in the complex plane starting at  formula_5. In doing so, one finds that the value of the extended function at a point formula_6 depends on the chosen curve from formula_5 to formula_8; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function. For example, let formula_9 be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of formula_10 in the complex plane, and then further along curves starting at formula_10, so that the values along a given curve vary continuously from formula_12. Extending to negative real numbers , one gets two opposite values for the square root—for example for —depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for th roots, logarithms, and inverse trigonometric functions. To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost  of possible value changes when one follows a closed path (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function formula_1 as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to formula_1. Examples. 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 the "principal branch" of the function. Set-valued analysis. Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology. Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied. Much of set-valued analysis arose through the study of mathematical economics and optimal control, partly as a generalization of convex analysis; the term "variational analysis" is used  by authors such as R. Tyrrell Rockafellar and Roger J-B Wets, Jonathan Borwein and Adrian Lewis, and Boris Mordukhovich. In optimization theory, the convergence of approximating subdifferentials to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point. There exist set-valued extensions of the following concepts from point-valued analysis: continuity, differentiation, integration, implicit function theorem, contraction mappings, measure theory, fixed-point theorems, optimization, and topological degree theory. Equations are generalized to inclusions. Types of multivalued functions. One can distinguish multiple concepts generalizing continuity, such as the closed graph property and upper and lower  hemicontinuity. There are also various generalizations of measure to multifunctions. Applications. Multifunctions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed-point theorem for multifunctions has been applied to prove existence of Nash equilibria (in the context of game theory, a multivalued function is usually referred to as a "correspondence"). This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity. Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the Michael selection  theorem, which provides another characterisation of paracompact spaces. Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski and Ryll-Nardzewski measurable selection theorem, Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in optimal control and the theory of differential inclusions. In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystals and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of  gauge field structures in many branches of physics.