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Multivalued function

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	Topics Multivalued function Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , a multivalued function (shortly: multifunction, other names: set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a total relationTotal relation In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is... ; i.e. every inputInput Input is the term denoting either an entrance or changes which are inserted into a system and which activate/modify a proces... is associated with one or more outputOutput Output is the term denoting either an exit or changes which exits a system and which activate/modify a process.... s. Strictly speaking, a "well-defined" functionFunction (mathematics) Overview In mathematics, a function relates each of its inputs to exactly one output.... associates one, and only one, outputFacts About Output Output is the term denoting either an exit or changes which exits a system and which activate/modify a process.... to any particular inputInput Overview Input is the term denoting either an entrance or changes which are inserted into a system and which activate/modify a proces... . The term "multivalued function" is, therefore, a misnomerMisnomer A misnomer is the wrong name or term for something; a misleading name, often idiomatic.... : true functions are single-valued. However, a multivalued function from A to B can be represented as a single-valued function from A to the set Power set In mathematics, given a set S, the power set of S, written , P or 2S, is the set of all subsets of S.... of nonempty subsets of B. Examples Each realReal number Overview In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... or complex numberComplex number In mathematics, a complex number is a number of the form ... except 0 has two square rootSquare root In mathematics, a square root of a number x is a number whose square is x.... s. The square roots of 4 are in the set . The square roots of 0 are described by the multisetMultiset In mathematics, a multiset differs from a set in that each member has a multiplicity, which is a natural number indicati... , because 0 is a root of multiplicityMultiplicity In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has.... 2 of the polynomial x². Each complex number has three cube rootCube root In mathematics, the cube rootof a number is the number which, when cubed, gives the original number.... s. The complex logarithmComplex logarithm One may define the natural logarithm also for all non-zero complex numbers z, but it is usually denoted log for mostly two rea... function is multiple-valued. The values assumed by log(1) are for all integerInteger Overview The integers consist of the positive natural numbers , their negatives and the number zero.... s . Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have Consequently arctan(1) may be thought of as having multiple values: π/4, 5π/4, −3π/4, and so on. This can be overcome by limiting the domain of tan(x) to -π/2 < x < π/2. Thus, the range of arctan(y) becomes -π/2 < y < π/2. These values from a limited domain are called principal valuePrincipal value ----In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values... s. The indefinite integral is a multivalued function of another function f. Its domain X is a set of functions. For any input f, it yields infinitely many possible solutions (the antiderivativeFacts About Antiderivative In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose... s of f). Notice that all of these examples refer to quasi-inverses of information-losing functions (i.e. imperfect inversesInverse function In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function.... of non-injective functions). Multivalued functions of a complex variable have branch pointBranch point In complex analysis, a branch point may be thought of informally as a point z0 at which a "multiple-valued function" cha... s. For example 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 which 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. Riemann surfaces A more sophisticated viewpoint replaces "multivalued functions" with functions whose domain is a Riemann surfaceRiemann surface In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional comp... (so named in honor of Bernhard RiemannBernhard Riemann Georg Friedrich Bernhard Riemann was a German mathematician who made important contributions to analysis and differential ... ). Types of multivalued functions One can differentiate many continuity concepts, primarily closed graph property and upper and lower hemicontinuityFacts About Hemicontinuity In mathematics, the concept of continuity as it is defined for single-valued functions is not immediately extendible to multi-valu... . (One should be warned that often the terms upper and lower semicontinuous are used instead of upper and lower hemicontinuous reserved for the case of weak topology in domain; yet we arrive at the collision with the reserved names for upper and lower simicontinuous real-valued function). There exist also various definitions for measurability of multifunction. History The practice of allowing function in mathematics to mean also multivalued function dropped out of usage at some point in the first half of the twentieth century. Some evolution can be seen in different editions of Course of Pure Mathematics by G. H. HardyG. H. Hardy Professor Godfrey Harold Hardy FRS was a prominent English mathematician, known for his achievements in number theory and ma... , for example. It probably persisted longest in the theory of special functions, for its occasional convenience. The theory of multivalued functions was fairly systematically developed for the first time in C. Berge,,Topological spaces" 1963. In physics, multivalued functions play an increasingly important role. They form the mathematical basis for DiracPaul Dirac Paul Adrien Maurice Dirac, OM, FRS was a British theoretical physicist and a founder of the field of quantum physics.... 's magnetic monopoleMagnetic monopole Overview In physics, a magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole".... s, for the theory of defectCrystallographic defect Crystalline solids have a very regular atomic structure: that is, the local positions of atoms with respect to each other are repe... s in crystal and the resulting plasticityPlasticity Plasticity has five meanings in the sciences:... of materials, for vorticesVortex A vortex is a spinning, often turbulent, flow with closed streamlines.... in superfluidSuperfluid Superfluidity is a phase of matter characterised by the complete absence of viscosity.... s and superconductors, and for phase transitionPhase transition In physics, a phase transition or phase change is the transformation of a thermodynamic system from one phase to anoth... s in these systems, for instance meltingMelting In physics, mellting is the process of heating a solid substance to a point where it turns liquid.... and quark confinement. They are the origin of gauge field structures in many branches of physics. Applications Multifunctions arise in optimal control theoryOptimal control Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimizatio... , especially differential inclusionDifferential inclusion In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form... s and related subjects as game theoryGame theory Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions ... , where the Kakutani fixed point theoremKakutani fixed point theorem Summary In mathematics, the Kakutani fixed point theorem is a fixed-point theorem that generalizes the Brouwer fixed point theorem.... for multifunctions has been applied to prove existence of Nash equilibriaNash equilibrium In game theory, the Nash equilibrium is a kind of solution concept of a game involving two or more players, where no player... . This amongst 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 hemicontinuous multifunctions usually possess continuous selections as stated in the Michael selection theoremMichael selection theorem In functional analysis, a branch of mathematics, the most popular version Michael selection theorem states the following: ... which provides another characterisation of paracompact spaces (see: E. Michael, Continuous selections I" Ann. of Math. (2) 63 (1956), and D. Repovs, P.V. Semenov, Ernest Michael and theory of continuous selections" arXiv:0803.4473v1). Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski—Ryll-Nardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in optimal controlOptimal control Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimizatio... and the theory of differential inclusionDifferential inclusion In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form... s. See also partial functionPartial function Overview In mathematics, a partial function is a relation that associates each element of a set with at most one element of anot... correspondenceCorrespondence (mathematics) In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings....