= Riemann integrability from almost full interval coverage
{c}
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 $\varepsilon/2$. The remaining length contributes at most $2M\delta$. Taking $\delta<\varepsilon/(4M)$ proves global <Riemann integrability> by the <Riemann integrability criterion>.
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