Definition:Negative Part/Also defined as/Negative Real Function
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Negative Part: Also defined as
Let $X$ be a set, and let $f: X \to \overline \R$ be an extended real-valued function.
Some sources define the negative part of an extended real-valued function $f$ as:
- $\forall x \in X: \map {f^-} x := \min \set {0, \map f x}$
That is:
- $\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$
Using this definition, the negative part is actually a negative function, which conforms to what feels more intuitively natural.
Also see
- Definition:Positive Part, the natural associate of negative part
- Results about negative parts can be found here.
Sources
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions