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

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