Definition:Integrable Unbounded Real Function/Also defined as

Definition

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

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

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

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

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