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Set-valued functions arise in [[Optimal control|optimal control theory]], especially [[Differential inclusion|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 equilibrium|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.<ref>{{cite journal |author=Ernest Michael |author-link=Ernest Michael |date=Mar 1956 |title=Continuous Selections. I |url=http://www.renyi.hu/~descript/papers/Michael_1.pdf |journal=Annals of Mathematics |series=Second Series |volume=63 |pages=361–382 |doi=10.2307/1969615 |jstor=1969615 |number=2 |hdl=10338.dmlcz/119700}}</ref><ref>{{cite journal |author1=Dušan Repovš |author1-link=Dušan Repovš |author2=P.V. Semenov |year=2008 |title=Ernest Michael and theory of continuous selections |journal=Topology Appl. |volume=155 |pages=755–763 |arxiv=0803.4473 |doi=10.1016/j.topol.2006.06.011 |number=8|s2cid=14509315 }}</ref> 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 inclusion|differential inclusions]].
== Notes ==
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