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:


 * $\forall x \in X: f^- \left({x}\right) := - \min \left\{{0, f \left({x}\right)}\right\}$

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

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

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