suggest globally quasi-convex function

Question

Can you suggest a function $f:R^2\to R, f\in C^2$, such that $f$ is globally quasi-convex (all its group sets are convex), but at no point convex?

Answer 1

The function $$f(x_1,x_2)=-\exp(x_1)$$ is nowhere convex, but all sublevel sets $\{(x_1,x_2):f(x_1,x_2)\le c\}$ are convex (being half-planes).