Existence of Integral on Union of Adjacent Intervals

Theorem
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.

Let $c$ be a point in $\left({a \,.\,.\, b}\right)$.

Then:


 * $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$


 * $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$.
 * $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$.

Necessary Condition
We need to prove that $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$.

To do this it suffices to show that for all $\epsilon > 0$, there exists a subdivision $S$ of $\left[{a \,.\,.\, b}\right]$ such that $U \left( {S} \right) – L \left( {S} \right) < \epsilon$.

Here, $U \left( {S} \right)$ and $L \left( {S} \right)$ are, respectively, the upper and lower sums of $f$ on $\left[{a \,.\,.\, b}\right]$ with respect to the subdivision $S$.

Let a strictly positive $\epsilon$ be given.

Since $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$, we know that a subdivision $S_1$ of $\left[{a \,.\,.\, c}\right]$ exists such that $U \left( {S_1} \right) – L \left( {S_1} \right) < \dfrac \epsilon 2$.

Since $f$ is Riemann integrable on $\left[{c \,.\,.\, b}\right]$, we know that a subdivision $S_2$ of $\left[{c \,.\,.\, b}\right]$ exists such that $U \left( {S_2} \right) – L \left( {S_2} \right) < \dfrac \epsilon 2$.

Define the subdivision $S = S_1 \cup S_2$.

We observe that $S$ is a subdivision of $\left[{a \,.\,.\, b}\right]$.

We get by the definition of upper sum:

Also, by the definition of lower sum:

These two equations give

This shows that $S$ satisfies $U \left( {S} \right) – L \left( {S} \right) < \epsilon$.

We conclude from this that $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$ since $S$ is a subdivision of $\left[{a \,.\,.\, b}\right]$.

Sufficient Condition
We need to prove that $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$.

Let a strictly positive $\epsilon$ be given.

The Riemann integrability of $f$ on $\left[{a \,.\,.\, b}\right]$ implies that a subdivision $P$ of $\left[{a \,.\,.\, b}\right]$ exists such that $U\left( {P} \right) – L\left( {P} \right) < \epsilon$.

Here, $U \left( {P} \right)$ and $L \left( {P} \right)$ are, respectively, the upper and lower sums of $f$ on $\left[{a \,.\,.\, b}\right]$ with respect to the subdivision $P$.

Define the subdivision $S = P \cup \left\{ {c} \right\}$.

We observe that $S$ equals $P$ if $c$ is a point in $P$, otherwise $S$ is a finer subdivision than $P$.

We have

This shows that $S$ satisfies $U\left( {S} \right) – L\left( {S} \right) < \epsilon$.

Define:


 * $S_1 = S \cap \left\{ {x: x \le c} \right\}$


 * $S_2 = S \cap \left\{ {x: x \ge c} \right\}$

We observe:


 * $S_1$ is a subdivision of $\left[{a \,.\,.\, c}\right]$.


 * $S_2$ is a subdivision of $\left[{c \,.\,.\, b}\right]$.


 * $S_1$ and $S_2$ are adjacent.


 * The union of $S_1$ and $S_2$ equals $S$.

We get by the definition of upper sum:

Also, by the definition of lower sum:

We have

This shows that $S_1$ satisfies $U\left( {S_1} \right) – L\left( {S_1} \right) < \epsilon$.

A similar deduction focusing on $S_2$ istead of $S_1$ shows that $S_2$ satisfies $U\left( {S_2} \right) – L\left( {S_2} \right) < \epsilon$.

$U\left( {S_1} \right) – L\left( {S_1} \right) < \epsilon$ gives that $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ since $S_1$ is a subdivision of $\left[{a \,.\,.\, c}\right]$.

$U\left( {S_2} \right) – L\left( {S_2} \right) < \epsilon$ gives that $f$ is Riemann integrable on $\left[{c \,.\,.\, b}\right]$ since $S_2$ is a subdivision of $\left[{c \,.\,.\, b}\right]$.