Definition:Negative Part

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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:

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

Also defined as

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.

Also see