Q:

Is there a function which satifies this condition?

Is there a function $f:\Bbb R\to\Bbb R$ which satisfies
$$\prod_{n=0}^{\infty} (f(x)^2-(2n+1)^2)=0$$
for all $x\in\Bbb R$? In other words, for all $x$, $f(x)$ is an odd integer.

A:

$$ y = 1+2\lfloor x\rfloor. \tag{1}$$

