> [!NOTE] Lemma > Let $f$ be a [[Real Function|real function]] on some [[Closed Real Interval|closed real interval]] $I$ and $a,b,c\in I.$ Then $f$ is [[Darboux Integrable Function|Darboux integrable]] on $[a,c]$ iff its integrable on $[a,b]$ and $[b,c]$ and in this case, $\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx $ **Proof**: ($\implies$) Suppose $f$ is integrable on $[a,c]$ Let $\varepsilon>0.$ By [[Necessary Condition for Darboux Integrability]], there exists a [[Finite Partition of Closed Real Interval|finite partition]] of $[a,b],$ say $P,$ such that the [[Upper Darboux Sum|upper]] and [[Lower Darboux Sum|lower Darboux sums]] of $f$ with respect to $P$ satisfy $U(f,P)-L(f,P)<\varepsilon.$Now choose a [[Refinement of Finite Partition of Closed Real Interval|refinement]] of $P$ that includes $b.$ We can regard the new partition as the union of a finite partition of $[a,b]$ and a finite partition of $[b,c].$ The upper sums for these partitions will add up to the upper sum for the refinement (and similarly the lower sums) $(\impliedby)$ ...