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

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

 \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?