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 $\omega(f;I)=sup _{x\in I} f(x)-inf _{x\in I} f(x)\ \in[0,\infty).$

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.

Proof sketch: On each subinterval, the upper contribution is $M_i\Delta x_i$ with $M_i=\sup f$ and the lower contribution is $m_i\Delta x_i$ with $m_i=\inf f$, so the difference is $(M_i-m_i)\Delta x_i=\omega(f;I_i)\Delta x_i$. The criterion is immediate from the definition of Riemann integrability as the ability to make $U(f,P)-L(f,P)$ arbitrarily small.