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set valued function wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 distinction from multivalued functions 2 example 3 set valued analysis 4 applications 5 notes 6 references 7 further reading 8 see also toggle the table of contents set valued function 2 languages 日本語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia function whose values are sets mathematics for multi valued functions of mathematical analysis see multivalued function for functions whose arguments are sets see set function this diagram represents a multi valued but not a proper single valued function because the element 3 in x is associated with two elements b and c in y a set valued function also called a correspondence or set valued relation is a mathematical function that maps elements from one set the domain of the function to subsets of another set 1 2 set valued functions are used in a variety of mathematical fields including optimization control theory and game theory set valued functions are also known as multivalued functions in some references 3 but this article and the article multivalued function follow the authors who make a distinction distinction from multivalued functions edit illustration distinguishing multivalued functions from set valued relations according to the criterion in page 29 of new developments in contact problems by wriggers and panatiotopoulos 2014 although other authors may distinguish them differently or not at all wriggers and panatiotopoulos 2014 distinguish multivalued functions from set valued functions which they called set valued relations by the fact that multivalued functions only take multiple values at finitely or denumerably many points and otherwise behave like a function 2 geometrically this means that the graph of a multivalued function is necessarily a line of zero area that doesn t loop while the graph of a set valued relation may contain solid filled areas or loops 2 alternatively a multivalued function is a set valued function f that has a further continuity property namely that the choice of an element in the set f x displaystyle f x defines a corresponding element in each set f y displaystyle f y for y close to x and thus defines locally an ordinary function example edit the argmax of a function is in general multivalued for example argmax x r cos x 2 π k k z displaystyle operatorname argmax _ x in mathbb r cos x 2 pi k mid k in mathbb z set valued analysis edit 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 4 implicit function theorem contraction mappings measure theory fixed point theorems 5 optimization and topological degree theory in particular equations are generalized to inclusions while differential equations are generalized to differential inclusions one can distinguish multiple concepts generalizing continuity such as the closed graph property and upper and lower hemicontinuity a there are also various generalizations of measure to multifunctions applications edit set valued functions arise in optimal control theory especially differential inclusions and related subjects as game theory where the kakutani fixed point theorem for set valued functions has been applied to prove existence of nash equilibria 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 6 7 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 notes edit some authors use the term semicontinuous instead of hemicontinuous references edit aliprantis charalambos d border kim c 2013 03 14 infinite dimensional analysis a hitchhiker s guide springer science business media p 523 isbn 978 3 662 03961 8 1 2 3 wriggers peter panatiotopoulos panagiotis 2014 05 04 new developments in contact problems springer p 29 isbn 978 3 7091 2496 3 repovš dušan 1998 continuous selections of multivalued mappings pavel vladimirovič semenov dordrecht kluwer academic isbn 0 7923 5277 7 oclc 39739641 aumann robert j 1965 integrals of set valued functions journal of mathematical analysis and applications 12 1 1 12 doi 10 1016 0022 247x 65 90049 1 kakutani shizuo 1941 a generalization of brouwer s fixed point theorem duke mathematical journal 8 3 457 459 doi 10 1215 s0012 7094 41 00838 4 ernest michael mar 1956 continuous selections i pdf annals of mathematics second series 63 2 361 382 doi 10 2307 1969615 hdl 10338 dmlcz 119700 jstor 1969615 archived from the original pdf on 2016 03 03 retrieved 2023 01 01 dušan repovš p v semenov 2008 ernest michael and theory of continuous selections topology appl 155 8 755 763 arxiv 0803 4473 doi 10 1016 j topol 2006 06 011 s2cid 14509315 further reading edit k deimling multivalued differential equations walter de gruyter 1992 c d aliprantis and k c border infinite dimensional analysis hitchhiker s guide springer verlag berlin heidelberg 2006 j andres and l górniewicz topological fixed point principles for boundary value problems kluwer academic publishers 2003 j p aubin and a cellina differential inclusions set valued maps and viability theory grundl der math wiss 264 springer verlag berlin 1984 j p aubin and h frankowska set valued analysis birkhäuser basel 1990 d repovš and p v semenov continuous selections of multivalued mappings kluwer academic publishers dordrecht 1998 e u tarafdar and m s r chowdhury topological methods for set valued nonlinear analysis world scientific singapore 2008 mitroi f c nikodem k wąsowicz s 2013 hermite hadamard inequalities for convex set valued functions demonstratio mathematica 46 4 655 662 doi 10 1515 dema 2013 0483 see also edit selection theorem ursescu theorem binary relation v t e function history list of specific functions types by domain codomain x 𝔹 𝔹 x 𝔹ⁿ 𝔹 x ℤ ℤ x x ℝ ℝ x ℝⁿ x x ℂ ℂ x ℂⁿ x classes properties constant identity linear polynomial rational algebraic analytic smooth continuous measurable injective surjective bijective constructions restriction composition λ inverse generalizations relation binary relation set valued multivalued partial implicit space higher order morphism functor category retrieved from https en wikipedia org w index php title set valued_function oldid 1335292844 categories variational analysis mathematical optimization control theory hidden categories articles with short description short description matches wikidata this page was last edited on 28 january 2026 at 12 13 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents set 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