---
tag: 5XW4N8KY3Y
kind: concept
name: Oscillation of a function
msc: [26A15]
sources:
  - url: https://en.wikipedia.org/wiki/Oscillation_(mathematics)
same_as:
  wikidata: Q7106411
  wikipedia: Oscillation (mathematics)
---

## Summary

The oscillation of a function at a point measures the size of the spread of its values near that point.
It is $0$ exactly at the points where the function is [[continuous|0EEE2CZB2A]], so it measures how badly continuity fails.
It is a basic tool in the study of the set of discontinuities, for example in [[Lebesgue's criterion for Riemann integrability|0565EGK6XP]].

## Explanations

### L1 human

Look at the values of a function on a small window around a point, and measure the gap between the largest and the smallest of them.
Shrink the window and see what this gap tends to.
That limit is the oscillation at the point.
For the sign function, the values near $0$ are $-1$, $0$ and $1$, so the oscillation at $0$ is $2$.