\begin{definition}[Definition:Positive Part]
Let $X$ be a set.
Let $f: X \to \overline \R$ be an extended real-valued function.
Then the '''positive part of $f$''', $f^+: X \to \overline \R$, is the extended real-valued function defined by:
:$\forall x \in X: \map {f^+} x := \max \set {0, \map f x}$
where the maximum is taken with respect to the extended real ordering.
That is:
:$\forall x \in X: \map {f^+} x := \begin {cases} \map f x & : \map f x \ge 0 \\ 0 & : \map f x < 0 \end {cases}$
\end{definition}