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  "body": "\n## Summary\n\nSet-valued analysis is the study of maps $F \\colon X \\rightrightarrows Y$ that assign to each point $x$ a subset $F(x)$ of $Y$, possibly empty or with many points.\nSuch maps are the [[multivalued functions|03CPD1HGF3]] of analysis, studied with tools of topology, convexity and differentiation.\nIts main notions are the graph and inverse of a set-valued map, its [[hemicontinuity|XNMKX7H4FD]], limits of sets, selections and derivatives of set-valued maps.\nSet-valued maps arise as solution sets of parametrized problems, as subdifferentials of nonsmooth functions, and as right-hand sides of [[differential inclusions|9ZSMF6NYKM]] in control theory.\nTypical results are the [[Kakutani fixed-point theorem|7V3HWGR32Z]] and the [[Michael selection theorem|3RJ1K36A9A]].\n\n## Explanations\n\n### L1 human\n\nAn ordinary function gives one answer for each input.\nMany questions in optimization and control have several answers, or none, for a given input.\nExamples are the set of minimizers of a problem, or the set of velocities a controlled system can choose.\nSet-valued analysis treats the whole set of answers as the value of a map.\nFor example, the map $F(a) = \\set{x \\in \\R : x^2 = a}$ gives two points for $a \u003e 0$, one point for $a = 0$ and the empty set for $a \u003c 0$.\nThe subject asks how such sets move when the input changes.\nIt also asks whether one can pick a single continuous answer from them, and how to differentiate them.\n",
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