\begin{definition}[Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below]
Let $f$ be a real function which is continuous everywhere.
Then the improper integral of $f$ over $\R$ is defined as:
:$\ds \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$
for some $c \in \R$.
Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.
\end{definition}