Theorem. Let $\displaystyle a,b,c \in \mathbb{R} $. Let $\displaystyle I $ be the smallest closed interval containing $\displaystyle a,b, $ and $\displaystyle c $ and suppose that $\displaystyle I \subseteq K \subseteq \mathbb{R} $. If $\displaystyle f: K \to \mathbb{R} $ is a function, then the integral $\displaystyle \smallint_{a}^{b} f $ exists if and only if the integrals $\displaystyle \smallint_{a}^{c} f $ and $\displaystyle \smallint_{b}^{c} f $ exist. So $\displaystyle \smallint_{a}^{b} f = \smallint_{a}^{c} f + \smallint_{c}^{b} f $.

Outline of Proof. (1) We want to show that $\displaystyle f $ is Riemann integrable on $\displaystyle [a,c] $ and $\displaystyle [c,b] $. Let $\displaystyle \epsilon >0 $. Try to find a corresponding $\displaystyle \delta $. Let $\displaystyle P_1 $ and $\displaystyle P_2 $ be partitions of $\displaystyle [a,c] $ and $\displaystyle [c,b] $ respectively with meshes less than $\displaystyle \delta $. Then look at $\displaystyle |\mathcal{R}(f,P_1)-\mathcal{R}_{2}(f,P_2)| $. (2) Show that any Riemann sum of $\displaystyle f $ on $\displaystyle P $ is within $\displaystyle \epsilon $ of $\displaystyle \smallint_{a}^{c} f + \smallint_{c}^{b} f $. We know that $\displaystyle f $ is bounded on $\displaystyle [a,b] $. Suppose $\displaystyle |f| \leq M $. Find a $\displaystyle \delta_1 $ corresponding to $\displaystyle \epsilon/3 $ on $\displaystyle [a,c] $ and $\displaystyle \delta_2 $ corresponding to $\displaystyle \epsilon/3 $ on $\displaystyle [c,b] $. Choose $\displaystyle \delta < \min(\delta_1, \delta_2, \eta) $ where $\displaystyle \eta $ is a function of $\displaystyle \epsilon $ and $\displaystyle M $. Now consider the following:

$\displaystyle \left|\mathcal{R}(f,P)- \left(\int_{a}^{c} f + \int_{c}^{b} f \right) \right| $

Here we have to be careful about the endpoints?


Is this in general correct? Any other faster ways of proving this?