Can a periodic function be represented with roots $= x^2$ where $x$ is an element of the integers?

Question

I only want the roots of the function to equal whole squares. So the function $\sin(\pi\times x)$ would not work, the roots can only be whole square numbers.

Answer 1

$$f(x) = \cases{\sin(\pi\sqrt{x}),& $x\ge 0$\cr x, & $x < 0$}$$ $$f(x) = 0\iff x\ge0, \sin(\pi\sqrt{x}) = 0\iff x\ge0,\pi\sqrt{x} = k\pi, k\in\Bbb Z\iff x = k^2, k\in\Bbb Z.$$ But the condition of periodicity is impossible.