Definition:Improper Integral/Unbounded Open Interval/Unbounded Below

Definition
Let $f$ be a real function which is continuous on the unbounded open interval $\openint {-\infty} b$.

Then the improper integral of $f$ over $\openint {-\infty} b$ is defined as:


 * $\ds \int_{\mathop \to -\infty}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$

for some $c \in \openint {-\infty} b$.