Definition:Improper Integral/Half Open Interval/Open Above

Definition

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:

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

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

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

When presenting an improper integral on a half open interval $\hointr a b$, 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 $\ds \int_a^{\mathop \to b} \map f t \rd t$.