Let $A$ and $B$ be closed sets (subsets of $\mathbb{R}^m$ and $\mathbb{R}^n$, say), and let $f : A \times B \rightarrow \mathbb{R}$ be a continuous function.

Consider the function $g : A \rightarrow B$ defined by

$g(x) = \underset{y \in B}{\operatorname{arg}\max} f(x,y)$

assuming some tie-breaking strategy for $f(x,y_1) = f(x,y_2)$. Clearly, $g$ may have discontinuities (but perhaps only countably many?).

If $A$ and $B$ are intervals of $\mathbb{R}$, $g$ corresponds to looking down the $y$-axis at the 3D graph of $f$, and marking the points on the "skyline".

Does this correspond to some known operation that has a established name?

My motivation is merely that I'm thinking about using this for a fun side project, so I'd like to know if it has a name and any known interesting properties beyond piecewise continuity(?). I've tagged this recreational accordingly; if it's an inappropriate question for MathOverflow, I apologize.