Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function on $\mathbb{R}^2$.
Is it true that, if $$\lim_{||(x,y)||\to{\infty}}{f(x,y)}=-{\infty}$$ then $f$ has a global maximum on $\mathbb{R}^2$?
I think that is true, but I don't know how to prove it. Can someone give me a proof or a counter-example?
It is true.
As $\lim_{||(x,y)||\to{\infty}}{f(x,y)}=-{\infty}$, there is $N>0$ such that if $\|(x,y)\|>N$, then $f(x,y)<-|f(0,0)|-1$.
Now, observe that $K:=\{(x,y)\in\Bbb R^2:\ \|(x,y)\|\le N\}$ is compact, so, since $f$ is continuous, Weierstrass' Theorem guarantees the existence of a maximum on $K$. Let $M:=\max_{x\in K} f(x)$.
If $f(0,0)\ge 0$, then $M\ge f(0,0)>-|f(0,0)|-1$, so $M$ is teh global Maximum of $f$.
If $f(0,0)<0$, then $M\ge f(0,0)=-|f(0,0)|>-|f(0,0)|-1$, and, again, $M$ is the global maximum of $f$.