\begin{definition}[Definition:Definite Integral/Darboux]
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a real function.
Let $f$ be bounded on $\closedint a b$.
Suppose that:
:$\ds \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$
where $\ds \underline {\int_a^b}$ and $\ds \overline {\int_a^b}$ denote the lower integral and upper integral, respectively.
Then the '''definite (Darboux) integral of $f$ over $\closedint a b$''' is defined as:
:$\ds \int_a^b \map f x \rd x = \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$
$f$ is formally defined as '''(properly) integrable over $\closedint a b$ in the sense of Darboux''', or '''(properly) Darboux integrable over $\closedint a b$'''.
More usually (and informally), we say:
:'''$f$ is (Darboux) integrable over $\closedint a b$.'''
\end{definition}