Oscillation of a function

$ f $ on a set $ E $

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

$$ \omega _ {E} ( f ) = \ \sup _ {P ^ \prime , P ^ {\prime\prime} \in E } \{ | f ( P ^ \prime ) - f ( P ^ {\prime\prime} ) | \} . $$

If the function is unbounded on $ E $, its oscillation on $ E $ is put equal to $ \infty $. For constant functions on $ E $( and only for these) the oscillation on $ E $ is zero. If the function $ f $ is defined on a subset $ E $ of $ \mathbf R ^ {n} $, then its oscillation at any point $ Q $ of the closure of $ E $ is defined by the formula

$$ \omega _ {Q , E } ( f ) = \ \inf _ {\begin{array}{c} U \\ Q \in U \end{array} } \omega _ {U \cap E } ( f ) , $$

where the infimum is taken over all neighbourhoods $ U $ of $ Q $. If $ Q \in E $, then in order that $ f $ be continuous at $ Q $ with respect to the set $ E $ it is necessary and sufficient that $ \omega _ {Q,E } ( f ) = 0 $.

Comments

The function $ Q \rightarrow \omega _ {Q,E } ( f ) $ is called the oscillation function of $ f $.

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