Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below

Definition
Let $f$ be a real function which is continuous everywhere.

Then the improper integral of $f$ over $\R$ is defined as:


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

for some $c \in \R$.

Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.