---
tag: D0R8455C26
kind: concept
name: Set-valued analysis
msc: [49J53, 54C60, 47H04]
sources:
  - url: https://en.wikipedia.org/wiki/Set-valued_function
---

## Summary

Set-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.
Such maps are the [[multivalued functions|03CPD1HGF3]] of analysis, studied with tools of topology, convexity and differentiation.
Its 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.
Set-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.
Typical results are the [[Kakutani fixed-point theorem|7V3HWGR32Z]] and the [[Michael selection theorem|3RJ1K36A9A]].

## Explanations

### L1 human

An ordinary function gives one answer for each input.
Many questions in optimization and control have several answers, or none, for a given input.
Examples are the set of minimizers of a problem, or the set of velocities a controlled system can choose.
Set-valued analysis treats the whole set of answers as the value of a map.
For example, the map $F(a) = \set{x \in \R : x^2 = a}$ gives two points for $a > 0$, one point for $a = 0$ and the empty set for $a < 0$.
The subject asks how such sets move when the input changes.
It also asks whether one can pick a single continuous answer from them, and how to differentiate them.