Definition:Improper Integral/Open Interval

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Definition

Let $f$ be a real function which is continuous on the open interval $\openint a b$.

Then the improper integral of $f$ over $\openint a b$ is defined as:

$\displaystyle \int_{\mathop \to a}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$

for some $c \in \openint a b$.


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 $c \in \openint a b$.

Thus:

$\openint a b = \hointl a c \cup \hointr c b$

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.