Bounded Function Continuous on Open Interval is Darboux Integrable

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Theorem

Let $f$ be a real function defined on an interval $\closedint a b$ such that $a < b$.

Let $f$ be continuous on $\openint a b$.

Let $f$ be bounded on $\closedint a b$.


Then $f$ is Darboux integrable on $\closedint a b$.


Proof

By Condition for Darboux Integrability, it suffices to show that, for a given strictly positive $\epsilon$, a subdivision $S$ of $\closedint a b$ exists such that:

$\map U S – \map L S < \epsilon$

where $\map U S$ and $\map L S$ are respectively the upper Darboux sum and lower Darboux sum of $f$ on $\closedint a b$ with respect to the subdivision $S$.


Since $f$ is bounded, a strictly positive bound $K$ exists for $f$ on $\closedint a b$.


Let a strictly positive $\epsilon$ be given, and choose a $\delta$ that satisfies:

$0 < \delta < \min \left({\dfrac \epsilon {6 K}, \dfrac {b - a} 2}\right)$


We have that $f$ is continuous on $\openint a b$.

As $\delta > 0$, $\closedint {a + \delta} {b - \delta}$ is a subset of $\openint a b$.

Thus $f$ is continuous on the interval $\closedint {a + \delta} {b - \delta}$.

By Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint {a + \delta} {b - \delta}$.


Since $f$ is Darboux integrable on $\closedint {a + \delta} {b - \delta}$, there exists a subdivision $S_\delta$ of $\closedint {a + \delta} {b - \delta}$ that satisfies:

$\map U {S_\delta} – \map L {S_\delta} < \dfrac \epsilon 3$

where $\map U {S_\delta}$ and $\map L {S_\delta}$ are respectively the upper Darboux sum and lower Darboux sum of $f$ on $\closedint {a + \delta} {b - \delta}$ with respect to the subdivision $S_\delta$.


Define the following subdivision of $\closedint a b$:

$S = S_\delta \cup \set {a, b}$


The upper Darboux sum of $f$ on $\closedint a b$ with respect to $S$ is per definition:

$\map U S = M_a \delta + \map U {S_\delta} + M_b \delta$

where:

$M_a$ is the supremum of $f$ on $\closedint a {a + \delta}$
$M_b$ is the supremum of $f$ on $\closedint {b - \delta} b$.

$M_a$ and $M_b$ exist by the least upper bound property of the real numbers because $f$ is bounded on $\closedint a {a + \delta}$ and $\closedint {b - \delta} b$.

The lower Darboux sum of $f$ on $\closedint a b$ with respect to $S$ is per definition:

$\map L S = m_a \delta + \map L {S_\delta} + m_b \delta$

where:

$m_a$ is the infimum of $f$ on $\closedint a {a + \delta}$
$m_b$ is the infimum of $f$ on $\closedint {b - \delta} b$

$m_a$ and $m_b$ exist by the Continuum Property, because $f$ is bounded on $\closedint a {a + \delta}$ and $\closedint {b - \delta} b$.


Define the sum:

\(\ds U'\) \(=\) \(\ds K \delta + \map U {S_\delta} + K \delta\)
\(\ds \) \(\ge\) \(\ds M_a \delta + \map U {S_\delta} + M_b \delta\) by $K \ge M_a$ and $K \ge M_b$
\(\ds \) \(=\) \(\ds \map U S\)

Define the sum:

\(\ds L'\) \(=\) \(\ds -K \delta + \map L {S_\delta} - K \delta\)
\(\ds \) \(\le\) \(\ds m_a \delta + \map L {S_\delta} + m_b \delta\) by $-K \le m_a$ and $-K \le m_b$
\(\ds \) \(=\) \(\ds \map L S\)

Therefore, $U'$ and $L'$ satisfy:

$U' \ge \map U S$
$L' \le \map L S$

From these two inequalities follows:

\(\ds \map U S – \map L S\) \(\le\) \(\ds U' - L'\)
\(\ds \) \(=\) \(\ds K \delta + \map U {S_\delta} + K \delta - \paren {-K \delta + \map L {S_\delta} -K \delta}\) Definitions of $U'$ and $L'$
\(\ds \) \(=\) \(\ds 4 K \delta + \map U {S_\delta} - \map L {S_\delta}\)
\(\ds \) \(<\) \(\ds 4 K \min \set {\dfrac \epsilon {6 K}, \dfrac {b - a} 2} + \map U {S_\delta} - \map L {S_\delta}\) by $\delta < \min \set {\dfrac \epsilon {6 K}, \dfrac {b - a} 2}$
\(\ds \) \(\le\) \(\ds 4 K \dfrac \epsilon {6 K} + \map U {S_\delta} - \map L {S_\delta}\) by $\min \set {\dfrac \epsilon {6 K}, \dfrac {b - a} 2} \le \dfrac \epsilon {6 K}$
\(\ds \) \(<\) \(\ds 4 K \dfrac \epsilon {6 K} + \dfrac \epsilon 3\) by $\map U {S_\delta} - \map L {S_\delta} < \dfrac \epsilon 3$
\(\ds \) \(=\) \(\ds \frac {2 \epsilon} 3 + \frac \epsilon 3\)
\(\ds \) \(=\) \(\ds \epsilon\)

Hence:

$\map U S – \map L S < \epsilon$

$\blacksquare$

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