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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.

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I'm baffled by the notation: what is "arg" here? Generally that is used for the argument of a complex number, but that can't be the case here. –  Robin Chapman Jul 1 '10 at 6:18
    
arg here means the argument (i.e the y) that achieves the max value. so g(x) is the value of y that maximizes f(x,y) –  Suresh Venkat Jul 1 '10 at 6:28
    
I think it's the "argument that maximizes"; that is, fix a value of x and ask which y leads to the largest value of f(x,y). –  Michael Burge Jul 1 '10 at 6:31
    
I believe this is standard notation in certain contexts (en.wikipedia.org/wiki/Arg_max , planetmath.org/encyclopedia/ArgMin.html). Thanks to Suresh and Michael Burge for restating my definition very clearly in terms that do not use arg max. –  Rahul Jul 1 '10 at 7:39
    
A lot will depend on your tie-breaking rule. If f is constant, you will have no problem finding nowhere continuous and even nonmeasurable functions g. –  Michael Greinecker Jul 1 '10 at 9:11
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1 Answer

up vote 1 down vote accepted

If you do not use any tie breaking strategy, you simply get the argmax-correspondence. That's the name I know. If B is compact, so maximizers actually exist, the argmax-correspondence has nonempty and compact values and is upper hemicontinuous. This follows from the Berge maximum theorem. You can find some material here, the result holds in a much more general context than given in the link. Upper hemicontinuous correspondences do not necessarily allow for continuous selections.

In the special case where the correspondence is single valued (for example if $f(x,\cdot)$ is always strictly concave), you actually get a continuous function, however.

It might be useful to know that the value function, the function $v:X\to\mathbb{R}$ given by $v(x)=f(x,y^* )$ with $y^*$ being in the argmax of $f(x,y)$, is always continuous.

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