Proof of Additivity of the Riemann Integral on Adjacent Intervals

Let $a,b,c\in\mathbb{R}$ with $a\le b\le c$, and let $f:[a,c]\to\mathbb{R}$ be Riemann integrable on $[a,c]$. We prove that the restrictions of $f$ to $[a,b]$ and $[b,c]$ are Riemann integrable and that

First we describe concatenation. Let

be a tagged partition of $[a,b]$, and let

be a tagged partition of $[b,c]$. Their concatenation is the tagged partition of $[a,c]$ obtained by joining the partition points and retaining the same tags on each subinterval. Its Riemann sum is exactly

Let

We first show that $f|_{[a,b]}$ is Riemann integrable on $[a,b]$. Fix $\varepsilon>0$. Since $f$ is Riemann integrable on $[a,c]$, there exists $\delta>0$ such that every tagged partition of $[a,c]$ of mesh less than $\delta$ has Riemann sum within $\varepsilon$ of $I$.

Choose once and for all a tagged partition $\mathcal{S}_0$ of $[b,c]$ with mesh less than $\delta$, and write

If $\mathcal{T}_1$ and $\mathcal{T}_2$ are tagged partitions of $[a,b]$ with mesh less than $\delta$, then the concatenations $\mathcal{T}_1\ast\mathcal{S}_0$ and $\mathcal{T}_2\ast\mathcal{S}_0$ are tagged partitions of $[a,c]$ with mesh less than $\delta$. Hence

Using the concatenation identity, we obtain

Thus the Riemann sums for $f|_{[a,b]}$ over sufficiently fine tagged partitions form a Cauchy family in $\mathbb{R}$, so they converge to some number $I_1\in\mathbb{R}$. It follows from the definition Riemann Integrability on a Closed Interval that $f|_{[a,b]}$ is Riemann integrable on $[a,b]$ and

The same argument, fixing a tagged partition of $[a,b]$, shows that $f|_{[b,c]}$ is Riemann integrable on $[b,c]$; write

Finally, let $\eta>0$. Choose tagged partitions $\mathcal{T}$ of $[a,b]$ and $\mathcal{S}$ of $[b,c]$ of sufficiently small mesh such that

and also the concatenation $\mathcal{T}\ast\mathcal{S}$ has Riemann sum within $\eta$ of $I$. Then

Since $\eta>0$ was arbitrary, one has $I=I_1+I_2$. Therefore

Subtracting $\int_a^b f(t)\,dt$ from both sides gives