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Let $f$ be a continuous function on $A$ which is a compact subset of $\mathbb{R}^2$. Due to Weierstrass theorem, I know that there exists $(a,b) \in A$ such that $f(a,b) = \max_{(x,y)\in A} f(x,y).$ This implies that the set $U = \arg \max_{(x,y)\in A} f(x,y)$ is non-empty. My question is the following: is it always possible to "choose" an element of U when U is non-singleton ?

I thought considering some lexicographical order on U, but I don't see how to prove that a maximal element always exists.

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    @Romain : If you mean the standard lexicographical ordering on $\mathbb{R}^2$, then $U$ is not necessarily compact in that order topology, but it is not difficult to show that $U$ has a maximal element in that ordering.2012-10-17

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