Definition:Improper Integral/Open Interval

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

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


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

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

Explanation
In this situation, there are two limits to consider.

The technique used here is to split the open interval into two half open intervals.

Let $c \in \left({a \,.\,.\, b}\right)$.

Thus:
 * $\left({a \,.\,.\, b}\right) = \left({a \,.\,.\, c}\right] \cup \left[{c \,.\,.\, b}\right)$

and use two improper integrals on half-open intervals.

The validity of this approach is justified by Sum of Integrals on Adjacent Intervals‎ for Continuous Functions.