## ::Classification of discontinuities

### ::concepts

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Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a **discontinuity** there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the **classification of discontinuities** in the simplest case of functions of a single real variable taking real values.

The oscillation of a function at a point quantifies these discontinuities as follows:

- in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
- in a jump discontinuity, the size of the jump is the oscillation (assuming that the value
*at*the point lies between these limits from the two sides); - in an essential discontinuity, oscillation measures the failure of a limit to exist.

**Classification of discontinuities sections**

Intro Classification The set of discontinuities of a function See also Notes References External links

PREVIOUS: Intro | NEXT: Classification |

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Function::''x'' Point::''l'' Cases::exist Limit::limits ''f''::equal Value::domain

{{#invoke:redirect hatnote|redirect}}
{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B=
{{#invoke:Message box|ambox}}
}}

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a **discontinuity** there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the **classification of discontinuities** in the simplest case of functions of a single real variable taking real values.

The oscillation of a function at a point quantifies these discontinuities as follows:

- in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
- in a jump discontinuity, the size of the jump is the oscillation (assuming that the value
*at*the point lies between these limits from the two sides); - in an essential discontinuity, oscillation measures the failure of a limit to exist.

**Classification of discontinuities sections**

Intro Classification The set of discontinuities of a function See also Notes References External links

PREVIOUS: Intro | NEXT: Classification |

<< | >> |