Definition:Improper Integral/Half Open Interval/Open Above

Definition
Let $f$ be a real function which is continuous on the half open interval $\left[{a \,.\,.\, b}\right)$.

Then the improper integral of $f$ over $\left[{a \,.\,.\, b}\right)$ is defined as:


 * $\displaystyle \int_a^{\mathop \to b} f \left({t}\right) \rd t := \lim_{\gamma \mathop \to b} \int_a^\gamma f \left({t}\right) \rd t$

Also denoted as
It is common to abuse notation and write:
 * $\displaystyle \int_a^b f \left({t}\right) \rd t$

which is understood to mean exactly the same thing as $\displaystyle \int_a^{\mathop \to b} f \left({t}\right) \rd t$.