# Definition:Improper Integral/Half Open Interval

## Definition

#### 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:

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

#### Open Below

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:

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