\begin{definition}[Definition:Improper Integral/Half Open Interval/Open Above]
Let $f$ be a real function which is continuous on the half open interval $\hointr a b$.
Then the improper integral of $f$ over $\hointr a b$ is defined as:
:$\ds \int_a^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to b} \int_a^\gamma \map f t \rd t$
\end{definition}