 Riemann integrability from almost full interval coverage

Let $|f|\le M$ on $[0,1]$. Suppose finitely many nonoverlapping closed intervals, of total length at least $1-\delta$, can be chosen for each $\delta >0$, with $f$ Riemann integrable on each. Refine the intervals to make their total Darboux sum gap less than $\epsilon /2$. The remaining length contributes at most $2M\delta$. Taking $\delta <\epsilon /(4M)$ proves global Riemann integrability by the Riemann integrability criterion.