# 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 $\omega _ {Q,E } ( f ) = 0$.

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