Definition:Improper Integral/Half Open Interval/Open Below

Definition

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:

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

The definition of an improper integral on a half open interval $\hointl a b$ can also be presented as:

$\ds \int_{\mathop \to a}^b \map f t \rd t := \lim_{\delta \mathop \to 0} \int_{a + \delta}^b \map f t \rd t$

When presenting an improper integral on a half open interval $\hointl a b$, it is common to abuse notation and write:

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

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