The Additivity Theorem

The Additivity Theorem for Riemann Integrable Functions: Let $f$ be a real-valued function on the interval $[a,b]$ , and let $c\in (a,b)$ . Then, $f$ is Riemann integrable on $[a,b]$ if and only if it is also Riemann integrable on $[a,c]$ and $[c,b]$ . In this case,

$\int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f$

Proof: Suppose that

\int _{a}^{c}f(x)\;d\alpha (x)=A

and

\int _{c}^{b}f(x)\;d\alpha (x)=B

for some

A,B\in \mathbb {R}

. Let

\epsilon >0

be given.

Since $\int _{a}^{c}f(x)\;d\alpha (x)=A$ we have that for $\epsilon _{1}={\frac {\epsilon }{2}}>0$ there exists a partition $P_{\epsilon _{1}}\in {\mathcal {P}}[a,c]$ such that for all partitions $P'\in {\mathcal {P}}[a,c]$ finer than $P_{\epsilon _{1}}$ , ( $P_{\epsilon _{1}}\subseteq P'$ ) and for any choice of $t_{k}$ 's in each $k^{\mathrm {th} }$ subinterval we have that:

{\begin{aligned}\quad \mid S(P',f,\alpha )-A\mid <\epsilon _{1}={\frac {\epsilon }{2}}\quad (*)\end{aligned}}

Similarly, since $\int _{c}^{b}f(x)\;d\alpha (x)=B$ we have that for $\epsilon _{2}={\frac {\epsilon }{2}}>0$ there exists a partition $P_{\epsilon _{2}}\in {\mathcal {P}}[c,b]$ such that for all partitions $P''\in {\mathcal {P}}[c,b]$ finer than $P_{\epsilon _{2}}$ , $(P_{\epsilon _{2}}\subseteq P''$ ) and for any choice of $u_{k}$ 's in each $k^{\mathrm {th} }$ subinterval we have that:

{\begin{aligned}\quad \mid S(P'',f,\alpha )-B\mid <\epsilon _{2}={\frac {\epsilon }{2}}\quad (**)\end{aligned}}

Let $P_{\epsilon }=P_{\epsilon _{1}}\cup P_{\epsilon _{2}}$ . Then $P_{\epsilon }$ is a partition of $[a,b]$ and for all partitions $P\in {\mathcal {P}}[a,b]$ finer than $P_{\epsilon }$ , ( $P_{\epsilon }\subseteq P$ ) we must have that $(*)$ and $(**)$ hold. Then for any choice of $v_{k}$ 's in each $k^{\mathrm {th} }$ subinterval we have that: