Existence of a saddle point Let f be a function of class $C^2$ on $ℝ^2$, each of whose critical points is isolated. Supposed that $A=\{x:f(x)>0\}$ is connected and contains two points of relative maximum. I want to show A contain a saddle point.
I have proved that there exist a number $\alpha>0$ such that $\{x:f(x)>\alpha\}$ is disconnected, but I do not know how to proceed.
This problem appears in a section of manifold of the book, how can I use the knowledge of manifold to prove it?

It's not true.  Counterexample: $$f(x,y) =  1 + \exp(-x)(x \exp(-x)+2 y^2 \exp(1-y^2) - 1)$$
Perhaps $A$  was supposed to be bounded? 
