Definition:Negative Part

Definition
Let $X$ be a set, and let $f: X \to \overline \R$ be an extended real-valued function.

Then the negative part of $f$, $f^-: X \to \overline \R$, is the extended real-valued function defined by:


 * $\forall x \in X: \map {f^-} x := -\min \set {0, \map f x}$

where the minimum is taken with respect to the extended real ordering.

That is:


 * $\forall x \in X: \map {f^-} x := \begin {cases} -\map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$

Hence the negative part of $f$ is actually defined as a positive function.

As a Real-Valued Function
Some sources insist that $f$ be a real-valued function instead.

However, $\R \subseteq \overline \R$ by definition of $\overline \R$.

Thus, the definition given above incorporates this approach.

As a Negative Function
Some sources define the negative part of $f$ 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.

In either case, utmost caution needs to be exercised in correct use of minus signs ("$-$") whenever dealing with the negative part.

Also see

 * Definition:Positive Part, the natural associate of negative part