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 always Riemann integrable.

Proof
As $$f$$ is continuous on $$\left[{a \,. \, . \, b}\right]$$, it follows from the Continuity Property that $$f$$ is bounded on $$\left[{a \,. \, . \, b}\right]$$.

Let $$P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\}$$ be a subdivision of $$\left[{a \,. \, . \, b}\right]$$.

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

That is, $$L \left({P}\right) = \sum_{k=1}^n m_{k} \left({x_{k} - x_{k - 1}}\right)$$ where $$m_k = \inf_{x \in \left[{x_{k - 1} \,. \, . \, x_{k}}\right]} f \left({x}\right)$$.

Let $$f$$ be bounded above on $$\left[{a \,. \, . \, b}\right]$$ by $$H$$.

Thus $$\forall t \in \left[{a \,. \, . \, b}\right]: f \left({t}\right) \le H$$.

In farticular, $$\forall 0 \le k \le n: m_k \le H$$. Then:

$$ $$ $$ $$ $$

Thus $$L \left({P}\right)$$ is bounded above by $$H \left({b - a}\right)$$.

Hence it has a supremum and we can define $$\int_a^b f \left({x}\right) dx = \sup_P L \left({P}\right)$$, where $$\sup_P$$ extends over all possible partitions of $$\left[{a \,. \, . \, b}\right]$$.

In the same way, we can consider the upper sum $$U \left({P}\right)$$ of $$f \left({x}\right)$$ on $$\left[{a \,. \, . \, b}\right]$$ belonging to the subdivision $$P$$.

Similarly, we can show that $$U \left({P}\right)$$ is bounded below and has an infimum.

After a little algebra, it is apparent that the use of the definition based on the upper sum would yield the result $$- \int_a^b -f \left({x}\right) dx = \inf_P U \left({P}\right)$$.

Now, if $$f$$ is continuous on $$\left[{a \,. \, . \, b}\right]$$, it follows from basic results in differentiation that $$F$$ is a primitive for $$f$$ on $$\left[{a \,. \, . \, b}\right]$$ iff $$-F$$ is a primitive for $$-f$$.

The result follows.