Oscillation of a function

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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 .


The function is called the oscillation function of .


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098