Continuous Real Function is Darboux Integrable

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,. \, . \, b}\right]$.

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

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

where $U \left({P}\right)$ and $L \left({P}\right)$ denote the upper sum and lower sum of $f \left({x}\right)$ on $\left[{a \,. \, . \, b}\right]$ belonging to the subdivision $P$.

Let $\epsilon > 0$.

A continuous function on a closed interval is uniformly continuous.

By the definition of uniform continuity, there exists a $\delta > 0$ such that if $x, y \in \left[{a \,. \, . \, b}\right]$ are such that $\left\vert{x – y}\right\vert < \delta$, then:
 * $\displaystyle \left\vert{ f \left({x}\right) – f \left({y}\right) }\right\vert < \frac {\epsilon} {b - a}$

Let $P = \left\{ {x_0, x_1, x_2, \ldots, x_n} \right\}$ be a subdivision of $\left[{a \,. \, . \, b}\right]$ such that:
 * $\displaystyle \max_{1 \le k \le n} \left({ x_k – x_{k-1} }\right) < \delta$

For all integers $k$ satisfying $1 \le k \le n$, it follows from the Heine-Borel theorem that $\left[{ x_{k-1} \,. \, . \, x_k }\right]$ is compact.

Because the image of a compact space by a continuous functions is compact, the image $f \left({ \left[{ x_{k-1} \, . \, . \, x_k }\right] }\right)$ is compact.

By the Heine-Borel theorem, $f \left({ \left[{ x_{k-1} \, . \, . \, x_k }\right] }\right)$ is closed.

The real number line is a complete metric space, so we can apply the fact the a subspace of a complete metric space is closed if and only if it is complete to conclude that $f \left({ \left[{ x_{k-1} \, . \, . \, x_k }\right] }\right)$ is complete.

Therefore, $f \left({ \left[{ x_{k-1} \, . \, . \, x_k }\right] }\right)$ contains its supremum and infimum.

Hence there exist $u_k, v_k \in \left[{ x_{k-1} \,. \, . \, x_k }\right]$ such that:
 * $\displaystyle f \left({u_k}\right) = \sup \left\{ {f \left({x}\right) : x \in \left[{ x_{k-1} \, . \, . \, x_k}\right]} \right\}$
 * $\displaystyle f \left({v_k}\right) = \inf \left\{ {f \left({x}\right) : x \in \left[{ x_{k-1} \, . \, . \, x_k}\right]} \right\}$

By assumption, $x_k – x_{k-1} < \delta$, so $\left\vert{u_k – v_k}\right\vert < \delta$.

It follows from the definition of $\delta$ that:
 * $\displaystyle f \left({u_k}\right) – f \left({v_k}\right) < \frac {\epsilon} {b – a}$

This gives:

as desired.