---
tag: XNMKX7H4FD
kind: concept
name: Hemicontinuity
msc: [54C60, "26E25"]
sources:
  - url: https://en.wikipedia.org/wiki/Hemicontinuity
same_as:
  wikidata: Q5711478
  wikipedia: Hemicontinuity
---

## Summary

Hemicontinuity is the notion of continuity for [[multivalued functions|03CPD1HGF3]], which assign to each point a set of values.
Upper hemicontinuity says that the value set cannot suddenly grow, and lower hemicontinuity says that it cannot suddenly shrink.
These notions appear in fixed-point theorems and in selection theorems such as the [[Michael selection theorem|3RJ1K36A9A]].
They also appear in economics and game theory, where best responses are often set-valued.

## Explanations

### L1 human

A multivalued function sends each point to a whole set, for example a set of best choices.
As the point moves a little, this set should change only a little.
Upper hemicontinuity forbids the set from exploding outward in a jump: near a point, the sets stay inside any open region around the set at that point.
Lower hemicontinuity forbids the set from losing parts in a jump: every part of the set at a point is still nearly reached by the sets at nearby points.