\begin{definition}[Definition:Negative Part]
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:
:$\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$
\end{definition}