Ruling out the existence of a strange polynomial

Question

Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that

$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$

and

$$\displaystyle \liminf_{(x,y) \in \mathbb{R}^2} f(x,y) = -\infty?$$

In other words, does there exist a polynomial $f$ which takes on positive values at every integer point, but still there exists a sequence $(x_k, y_k)$ of real pairs such that $\lim_{k \rightarrow \infty} f(x_k, y_k) = -\infty$?

Note that if such a sequence exists, the norm of its elements must tend to infinity. This is because $f$ is continuous, and therefore the image of any compact set under $f$ is necessarily compact, and thus in particular must be bounded.

Answer 1

The polynomial $f(x,y)=(x^2+1)(5y^2+5y+1)\in\mathbb{Z}[x,y]$ is an example. Note that $5y^2+5y+1>0$ for $y\in\mathbb{Z}$, but $5y^2+5y+1<0$ at $y=-\frac{1}{2}$.