\begin{definition}[Definition:Horizontal Section of Function]
Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be an extended real-valued function. 
Let $y \in Y$.
We define the '''$y$-horizontal section''' $f^y$ of $f$ by: 
:$\map {f^y} x = \map f {x, y}$
for each $x \in X$.
\end{definition}