Oscillation criterion lemma

Let $f:[a,b]\to\mathbb{R}$ be bounded . For a subinterval $I\subseteq[a,b]$, define the oscillation of $f$ on $I$ by $\omega(f;I)=\sup_{x\in I} f(x)-\inf_{x\in I} f(x)\ \in[0,\infty),$ where sup and inf denote the supremum and infimum .

Oscillation identity: If $P=\{a=x_0<x_1<\cdots<x_n=b\}$ is a partition , then $U(f,P)-L(f,P)=\sum_{i=1}^n \omega\bigl(f;[x_{i-1},x_i]\bigr)\,(x_i-x_{i-1}).$

Oscillation criterion (Riemann integrability): A bounded function $f$ is Riemann integrable on $[a,b]$ if and only if for every $\varepsilon>0$ there exists a partition $P$ such that $\sum_{i=1}^n \omega\bigl(f;[x_{i-1},x_i]\bigr)\,(x_i-x_{i-1})<\varepsilon.$

This criterion repackages $U(f,P)-L(f,P)$ into a concrete “local oscillation” quantity and is a standard starting point for proving integrability of classes of functions.