> [!NOTE] Lemma > Let $c\in \mathbb{R}$ and $f:x\mapsto c$ be a [[Real Function|real function]]. Let $a,b\in \mathbb{R}.$ Then $f$ is [[Riemann integration|integrable]] on $[a,b]$ and its integral is given by $\int_{a}^{b} f(x) \, dx = c(b-a).$ **Proof**: $f$ is monotone so by [[Monotone Real Function is Darboux Integrable]], $f$ is indeed integrable. Let $P_{n}$ be a finite partition of $[a,b]$ into $n$ equal intervals. STS to show that $U(f,P_{n})\to c(b-a)$ as $n\to \infty.$ Thus by [[Upper Darboux Sums of Darboux Integrable Function Converge to Darboux Integral as Mesh Size Approaches Zero]], $\int_{a}^{b} f(x) \, dx$ equals this limit. Proof: Clearly the upper and lower sums are constant and always equal $c(b-a).$