---
tag: 04R9K5H562
kind: concept
name: Classification of discontinuities
msc: [26A15]
uses: [0EEE2CZB2A, 0909PWZR62]
sources:
  - url: https://en.wikipedia.org/wiki/Classification_of_discontinuities
    locator: Classification
same_as:
  wikidata: Q541961
  wikipedia: Classification of discontinuities
---

## Summary

A point of the domain where a real function is not [[continuous|0EEE2CZB2A]] is a discontinuity. Discontinuities are sorted by the behavior of the [[one-sided limits|0909PWZR62]] at the point.

## Remarks

At a [[removable discontinuity|0QG9C579SR]], the limit exists but differs from the value.
At a [[jump discontinuity|05B0FKFXQA]], both one-sided limits exist but they differ.
At an essential discontinuity, at least one one-sided limit does not exist. An example is $\sin(1/x)$ at $0$, with any value at $0$.
A [[monotone function has countably many discontinuities|0WYTM4C9QG]], and all of them are jumps.

## Explanations

### L1 human

A function can fail to be continuous in different ways.
A hole in the graph can be repaired, a jump cannot, and some functions oscillate too wildly to have any limit.