Definition:Integrable Function/Unbounded

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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:

\(\ds \map {f^+} x\)

\(:=\)

\(\ds \max \set {0, \map f x}\)

\(\ds \map {f^-} x\)

\(:=\)

\(\ds -\min \set {0, \map f x}\)

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:

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

Also defined as

Sources which define the negative part of $f$ as negative real function:

$\map {f^-} x := \min \set {0, \map f x}$

consequently define the (definite) integral of $f$ as:

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

Sources

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