= Solution
A <Riemann integrable> function on a closed interval is bounded and has equal <upper and lower Darboux integrals>. More explicitly, for a <partition of an interval> $P:a=x_0<x_1<\cdots<x_m=b$, define the <Darboux sums>
$$
L(f,P)=\sum_{j=1}^m\inf_{[x_{j-1},x_j]}f\,(x_j-x_{j-1}),\qquad U(f,P)=\sum_{j=1}^m\sup_{[x_{j-1},x_j]}f\,(x_j-x_{j-1}).
$$
Then \b[<Riemann integrability> means]
$$
\boxed{\sup_P L(f,P)=\inf_P U(f,P),}
$$
and this common value is the <Riemann integral>. Equivalently, the <Riemann integrability criterion> requires that for every $\varepsilon>0$ some <partition of an interval> satisfies $U(f,P)-L(f,P)<\varepsilon$. Indeed, the common <partition refinement> of two partitions increases each lower <Darboux sum> and decreases each upper <Darboux sum>, so equality of the two extremal values is equivalent to arbitrarily small gaps.
Back to article page
