## ::Classification of discontinuities

### ::concepts

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

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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