Constructing A Type of Function

Question

Let $f(a,b)$ be a continuous function that is only an integer when $a$ and $b$ are both integers. What is one such definition of $f(a,b)$?

Answer 1

For instance, $$f(a,b)=\min\left\{\left(a-\left\lfloor a\right\rfloor\right)^2+\left(b-\left\lfloor b\right\rfloor\right)^2;\left(a-\left\lceil a\right\rceil\right)^2+\left(b-\left\lfloor b\right\rfloor\right)^2; \left(a-\left\lceil a\right\rceil\right)^2+\left(b-\left\lceil b\right\rceil\right)^2;\left(a-\left\lfloor a\right\rfloor\right)^2+\left(b-\left\lceil b\right\rceil\right)^2\right\}$$

It is continuous, and it satisfies $0\le f(a,b)\le \frac{1}2$ and $f(a,b)=0\iff (a,b)\in\Bbb Z^2$.