Definition:Negative Part/Also defined as/Negative Real Function

From ProofWiki
Jump to navigation Jump to search

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

  • Results about negative parts can be found here.


Sources