## Integral of a Sum

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?