Definition:Improper Integral/Unbounded Open Interval/Unbounded Below

Definition
Let $f$ be a real function which is continuous on the unbounded open interval $\left({-\infty \,.\,.\, b}\right)$.

Then the improper integral of $f$ over $\left({-\infty \,.\,.\, b}\right)$ is defined as:


 * $\displaystyle \int_{\mathop \to -\infty}^{\mathop \to b} f \left({t}\right) \ \mathrm d t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c f \left({t}\right) \ \mathrm d t + \lim_{\gamma \mathop \to b} \int_c^\gamma f \left({t}\right) \ \mathrm d t$

for some $c \in \left({-\infty \,.\,.\, b}\right)$.