# Oscillation of a function

From Encyclopedia of Mathematics

* on a set *

The difference between the least upper and the greatest lower bounds of the values of on . In other words, the oscillation of on is given by

If the function is unbounded on , its oscillation on is put equal to . For constant functions on (and only for these) the oscillation on is zero. If the function is defined on a subset of , then its oscillation at any point of the closure of is defined by the formula

where the infimum is taken over all neighbourhoods of . If , then in order that be continuous at with respect to the set it is necessary and sufficient that .

#### Comments

The function is called the oscillation function of .

#### References

[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 120 |

[a2] | R.R. Goldberg, "Methods of real analysis" , Blaisdell (1964) pp. 129 |

**How to Cite This Entry:**

Oscillation of a function. A.A. Konyushkov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Oscillation_of_a_function&oldid=12236

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098