Definition:Negative Part/Also defined as

Negative Part: Also defined as

This page presents variants of the definition of the negative part of an extended real-valued function.

Some sources insist, when defining the negative part, that $f$ be a real-valued function:

$f: X \to \R$

That is, that the codomain of $f$ includes neither the positive infinity $+\infty$ nor the negative infinity $-\infty$.

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

Hence it is still the case that:

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

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.

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