I'm searching for a function $f: \Bbb R^2 \rightarrow \Bbb R$ which has following properties:
- Both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are continuous in $\Bbb R^2$.
- There are infinitely many critical points* , in all of which, $f$ has local maximum.
- $f$ is unbounded both from above and from below.
Could you give me some hints, or even better, show me explicitly the desired function?
*$(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$.