Definition:Integrable Function/Unbounded

Definition
Let $f: \R \to \R$ be a real function.

Let $f$ be unbounded on the open interval $\openint a b$.

Let:
 * $f^+$ denote the positive part of $f$
 * $f^-$ denote the negative part of $f$,

that is:
 * $\map {f^+} x := \max \set {0, \map f x}$
 * $\map {f^-} x := \min \set {0, \map f x}$

Let $f^+$ and $-f^-$ both be integrable on $\openint a b$.

Then $f$ is '''integrable on $\openint a b$ and its (definite) integral is understood to be:


 * $\ds \int_a^b \map f x \rd x := \int_a^b \map {f^+} x \rd x - \int_a^b \paren {-\map {f^-} x} \rd x$