> [!NOTE] Theorem > Let $[a,b]$ be a [[Closed Real Interval|closed real interval]]. Let $f,g:[a,b]\to \mathbb{R}$ be [[Darboux Integrable Function|Darboux integrable]]. Then $fg:[a,b]\to \mathbb{R}$ is also Darboux integrable. **Proof**: We have $fg = \frac{1}{2} ((f+g)^{2}-f^{2}-g^{2}).$It follows from [[Linearity of Darboux Integral]] that $f+g$ is Darboux integrable. It follows from [[Continuous Real Function of Darboux Integrable Function is Darboux Integrable]] that $f^{2},$ $g^{2}$ and $(f+g)^{2}$ are Darboux integrable. It follows from linearity again that $fg$ is integrable.