Oscillation of a function

Let $f:E\to\mathbb{R}$ be a bounded function and let $A\subseteq E$ be nonempty. The oscillation of $f$ on $A$ is

using the supremum and infimum .

For Riemann integration , oscillation on subintervals controls the gap between upper and lower sums : on an interval $I$, the contribution to $U(f,P)-L(f,P)$ is the oscillation on $I$ times the interval length.

Examples:

• If $f$ is constant on $A$, then $\operatorname{osc}(f;A)=0$.
• For $f(x)=x$ on $A=[0,1]$, $\operatorname{osc}(f;A)=1-0=1$.
• For $f=\mathbf{1}_{\mathbb{Q}}$ on any nontrivial interval $I\subset\mathbb{R}$, $\operatorname{osc}(f;I)=1-0=1$.