\begin{definition}[Definition:Integrable Function/Unbounded Above]
Let $f: \R \to \R$ be a positive real function.
Let $f$ be unbounded above on the open interval $\openint a b$.
Let $f_m$ denote the function $f$ truncated by $m$ for $m \in \Z_{>0}$.
Suppose that $f_m$ is Darboux integrable on $\openint a b$ for all $m \in \Z_{>0}$.
Suppose also that the following limit exists:
:$\ds \lim_{m \mathop \to \infty} \int_a^b \map {f_m} x \rd x$
Then $f$ is '''integrable on $\openint a b$''' and can be expressed in the conventional notation of the Darboux integral:
:$\ds \int_a^b \map f x \rd x$
\end{definition}