Definition:Improper Integral/Half Open Interval/Open Below

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Let $f$ be a real function which is continuous on the half open interval $\hointl a b$.

Then the improper integral of $f$ over $\hointl a b$ is defined as:

$\displaystyle \int_{\mathop \to a}^b \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^b \map f t \rd t$

Also denoted as

It is common to abuse notation and write:

$\displaystyle \int_a^b \map f t \rd t$

which is understood to mean exactly the same thing as $\displaystyle \int_{\mathop \to a}^b \map f t \rd t$.