analysis - An inequality about a continuous function.

Let $\Omega \subset \Bbb R ^2$ be bounded and closed, and let $g : \Omega \to \Bbb [0, \infty)$ be continuous. Let $g(x_0,y_0)=\max \limits_{\Omega} g(x,y)$. Show that:

$\exists \rho_{x_0},\rho_{y_0}>0$, $\forall(x,y)\in \Omega,|x-x_0|<\rho_{x_0},|y-y_0|<\rho_{y_0}$, then $g(x,y_0)\geq g(x,y)$.

In fact, I think the $\Omega$ is bounded and closed is not necessary. All of the above is my guess, I really don't know whether it is right. Thanks for any answer or advice.

Answer

It is not true. The strict inequality is trivially false by taking $x=x_0$, $y=y_0$. The non-strict one is not true either. Consider $g(x,y)=-(x-y)^2$ and $(x_0,y_0)=(0,0)$. Then
$$
g(x,y_0)=-x^2$$
for all $x\ne x_0$.