Positive Part of Vertical Section of Function is Vertical Section of Positive Part

Theorem

Let $X$ and $Y$ be sets.

Let $f : X \times Y \to \overline \R$ be a function.

Let $x \in X$.

Then:

$\paren {f_x}^+ = \paren {f^+}_x$

where:

$f_x$ denotes the $x$-vertical function of $f$

$f^+$ denotes the positive part of $f$.

Proof

Fix $x \in X$.

Then, we have, for each $y \in Y$:

\(\ds \map {\paren {f^+}_x} y\)

\(=\)

\(\ds \map {f^+} {x, y}\)

\(\ds \)

\(=\)

\(\ds \max \set {0, \map f {x, y} }\)

Definition of Positive Part

\(\ds \)

\(=\)

\(\ds \max \set {0, \map {f_x} y}\)

Definition of Vertical Section of Function

\(\ds \)

\(=\)

\(\ds \map {\paren {f_x}^+} y\)

Definition of Positive Part

$\blacksquare$