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 $f$.

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:

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

1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions