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

Negative Part: Also defined as
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 $\overline \R$.

Thus, the main definition as provided on incorporates this approach.

Hence it is still the case that:
 * $\forall x \in X: \map {f^-} x := \begin {cases} -\map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$

Also see

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