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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: IntroNEXT: Classification
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{{#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: IntroNEXT: Classification
<<>>