---
tag: RN4G8SGP6P
kind: concept
name: Semicontinuity
msc: [54C08, 26A15]
sources:
  - url: https://en.wikipedia.org/wiki/Semi-continuity
same_as:
  wikidata: Q955072
  wikipedia: Semi-continuity
---

## Summary

Semicontinuity is a one-sided form of [[continuity|0VGFHTM75Y]] for functions with values in the [[extended real line|0GSZ7B0CGR]].
A function is upper semicontinuous when its value at a point may lie above the nearby values but not below them.
It is lower semicontinuous in the opposite case.
A function is continuous exactly when it is both.
Lower semicontinuous functions attain their minimum on nonempty [[compact spaces|0TVJZYB6SH]], which makes them basic in optimisation and the calculus of variations.

## Explanations

### L1 human

Think of the graph of a function with a jump.
At an upper semicontinuous point, the value is at least as high as the nearby values tend to be.
The function may drop sharply as you move away from the point, but it cannot rise sharply.
The [[floor function|08GR54CKEA]] is upper semicontinuous: at an integer $n$ its value is $n$, and just to the left it is $n - 1$.
The [[indicator function|0RCQZ36MME]] of an open set is lower semicontinuous, and the indicator function of a closed set is upper semicontinuous.