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:


 * if $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$ and $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$

and


 * If $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$, $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$ and $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$

The case $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$
We start by proving 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 $U\left( P \right) – L\left( P \right) < \epsilon$ where $U\left( P \right)$ 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

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

Define the subdivisions $S_1 = S \setminus \left\{ x: x>c \right\}$ and $S_2 = S \setminus \left\{ x: x<c \right\}$.

We observe:


 * $S_1$ is a subdivision of $\left[{a \,.\,.\, c}\right]$, and $S_2$ is a subdivision of $\left[{c \,.\,.\, b}\right]$.


 * $S_1$ and $S_2$ are adjacent as they both contain the point $c$.


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

Since $S_1$ and $S_2$ are adjacent and $S = S_1 \cup S_2$, we get by the definition of upper sum:


 * $U \left( S \right) = U \left( S_1 \right) + U \left( S_2 \right)$

and by the definition of lower sum:


 * $L \left(S \right) = L \left(S_1 \right) + L \left(S_2 \right)$

We have

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

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

What remains to prove is that $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ equals $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$.

We have


 * $U\left( S \right)$ approaches $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ as $\epsilon$ approaches 0 since $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$, the interval of which $S$ is a subdivision


 * $U\left( S_1 \right)$ approaches $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x$ as $\epsilon$ approaches 0 since $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$, the interval of which $S_1$ is a subdivision


 * $U\left( S_2 \right)$ approaches $\displaystyle \int_c^b f \left({x}\right) \ \mathrm d x$ as $\epsilon$ approaches 0 since $f$ is Riemann integrable on $\left[{c \,.\,.\, b}\right]$, the interval of which $S_2$ is a subdivision

Combination Theorem for Sequences/Sum Rule applied to the two last limit expressions above gives that the limit of $U\left(S_1\right) + U\left(S_2\right)$ exists as well, and that it satsifies:


 * $U\left(S_1\right) + U\left(S_2\right)$ approaches $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ as $\epsilon$ approaches 0

Furthermore, since $U\left(S\right) = U\left(S_1\right) + U\left(S_2\right)$:


 * $U\left( S \right)$ approaches $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$ as $\epsilon$ approaches 0

Since $U\left( S \right)$ approaches $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ also as ϵ approaches 0, we conclude that $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ equals $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$.

The case $f$ is Riemann integrable on $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$
We start by proving that $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$.

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

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 $U \left( S_1 \right) – L \left( S_1 \right) < \dfrac \epsilon 2$ where $U \left( S_1 \right) $ and $L \left( S_1 \right) $ are, respectively, the upper and lower sums of $f$ on $\left[{a \,.\,.\, c}\right]$ with respect to the subdivision $S_1$.

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 $U \left( S_2 \right) – L \left( S_2 \right) < \dfrac \epsilon 2$ where $U \left( S_2 \right) $ and $L \left( S_2 \right) $ are, respectively, the upper and lower sums of $f$ on $\left[{c \,.\,.\, b}\right]$ with respect to the subdivision $S_2$.

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

We observe that $S$ is a subdivision of $\left[{a \,.\,.\, b}\right]$, the union of $\left[{a \,.\,.\, c}\right]$, the interval of which $S_1$ is a subdivision, and $\left[{c \,.\,.\, b}\right]$, the interval of which $S_2$ is a subdivision.

Since $\left[{a \,.\,.\, c}\right]$ and $\left[{c \,.\,.\, b}\right]$ are adjacent, the equation $S = S_1 \cup S_2$ implies by the definition of upper sum:


 * $U(S) = U(S_1) + U(S_2)$

and by the definition of lower sum:


 * $L(S) = L(S_1) + L(S_2)$

These two equations give

which shows that $S$ satisfies $U \left( S\right) – L \left( S \right) < \epsilon$, from which we conclude that $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$, the interval of which $S$ is a subdivision.

What remains to prove is that $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x$ equals $\displaystyle \int_a^c f \left({x}\right) \ \mathrm d x + \int_c^b f \left({x}\right) \ \mathrm d x$.

The proof of this is word by word the same as the proof of the same statement in the previous case, and this completes the proof of the theorem.