Oscillation - Imported Knowlpedia Content

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

\operatorname{osc}(f;A)=\sup\{|f(x)-f(y)|:x,y\in A\}.

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

Remarks

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 $\operatorname{osc}(f;I)=1$ .