In order to show the existence of Nash equilibrium in a continuous game for concave objective functions (Theorem.1. (Rosen 1965) ), i found a result that says that the function $\Gamma $ is upper semicontinuous, where :
$\Gamma(x)=\{y \in E|\rho(x,y)=\displaystyle\max_{z\in E}\rho(x,z)\}$
Here we note that:
$\bullet\:\Gamma : E\rightarrow P(E)$
$\bullet\:E=\displaystyle\Pi_{i\in N}^{n}E_{i}\:\:$ avec $E_{i}\subset\mathbb{R}$ is convex, compact $\forall i\in \{1,...n\}$
$\bullet\:\rho : E\times E\rightarrow \mathbb{R}$ is concave in $y$ for each fixed $x \in E$
$\bullet\:\rho$ is continuous in $E\times E$
So my question is:
How to show that $\Gamma$ is upper semicontinuous ?