Theorem: Bounded Function Continuous on Open Interval is Darboux Integrable

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 $\closedint a b$ {{WRT}} 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:
:$(1): \quad 0 < \delta < \min \set {\dfrac \epsilon {6 K}, \dfrac {b - a} 2}$
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:
:$(2): \quad \map U {S_\delta} - \map L {S_\delta} < \dfrac \epsilon 3$
where $\map U {S_\delta}$ and $\map L {S_\delta}$ are respectively the upper Darboux sum and lower Darboux sum of $f$ on $\closedint {a + \delta} {b - \delta}$ {{WRT}} 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 $\closedint a b$ {{WRT}} $S$ 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 Least Upper Bound Property because $f$ is bounded on $\closedint a {a + \delta}$ and $\closedint {b - \delta} b$.
The lower Darboux sum of $f$ on $\closedint a b$ {{WRT}} $S$ 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:
{{begin-eqn}}
{{eqn | l = U'
      | r = K \delta + \map U {S_\delta} + K \delta
}}
{{eqn | o = \ge
      | r = M_a \delta + \map U {S_\delta} + M_b \delta
      | c = by $K \ge M_a$ and $K \ge M_b$
}}
{{eqn | r = \map U S
}}
{{end-eqn}}
Define the sum:
{{begin-eqn}}
{{eqn | l = L'
      | r = -K \delta + \map L {S_\delta} - K \delta
}}
{{eqn | o = \le
      | r = m_a \delta + \map L {S_\delta} + m_b \delta
      | c = by $-K \le m_a$ and $-K \le m_b$
}}
{{eqn | r = \map L S
}}
{{end-eqn}}
Therefore, $U'$ and $L'$ satisfy:
:$U' \ge \map U S$
:$L' \le \map L S$
From these two inequalities follows:
{{begin-eqn}}
{{eqn | l = \map U S - \map L S
      | o = \le
      | r = U' - L'
}}
{{eqn | r = K \delta + \map U {S_\delta} + K \delta - \paren {-K \delta + \map L {S_\delta} -K \delta}
      | c = Definitions of $U'$ and $L'$
}}
{{eqn | r = 4 K \delta + \map U {S_\delta} - \map L {S_\delta}
}}
{{eqn | o = <
      | r = 4 K \min \set {\frac \epsilon {6 K}, \frac {b - a} 2} + \map U {S_\delta} - \map L {S_\delta}
      | c = by $(1)$
}}
{{eqn | o = \le
      | r = 4 K \frac \epsilon {6 K} + \map U {S_\delta} - \map L {S_\delta}
      | c = by $\min \set {\dfrac \epsilon {6 K}, \dfrac {b - a} 2} \le \dfrac \epsilon {6 K}$
}}
{{eqn | o = <
      | r = 4 K \frac \epsilon {6 K} + \frac \epsilon 3
      | c = by $(2)$
}}
{{eqn | r = \frac {2 \epsilon} 3 + \frac \epsilon 3
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
Hence:
:$\map U S - \map L S < \epsilon$
{{qed}}
Category:Integral Calculus
Category:Bounded Real-Valued Functions
Category:Continuous Real Functions
Category:Darboux Integrable Functions
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