Oscillation

An oscillation of a bounded function $f$ on a set $A\subseteq\mathbb R$ is the number

If $A$ is an interval, this equals $\sup\{f(x):x\in A\}-\inf\{f(x):x\in A\}$.

Oscillation is used in the oscillation criterion for Riemann integrability , and it also detects continuity: $x$ is a discontinuity point exactly when the oscillation over shrinking intervals around $x$ fails to go to $0$.

Examples:

• For $f(x)=x$ on $[0,1]$, one has $\operatorname{osc}(f;[0,1])=1$.
• If $f$ is the indicator function of $\mathbb Q\cap[a,b]$, then on every nontrivial subinterval $I\subseteq[a,b]$ one has $\operatorname{osc}(f;I)=1$.