An elementary problem concerning real nuumbers

Question

Take any $y\in \mathbb{R}$, $r>0$ and let $I_{y,r}=(y,y+r)$. Consider the set $A=\bigcup\limits_{i=1}^\infty (i^2,i^2+1)\cup (-i^2,-i^2+1)$, how to explicitly write the set $A\cap I_{y,r}$ as a union open disjoint intervals?

Answer 1

The main part is $$\bigcup_{k\in\Bbb Z, k\ne0,\\ y< k|k|< y+r-1}(k|k|,k|k|+1).$$ If $|\lfloor y\rfloor|$ is a non-zero perfect square, add $$(y,\min\{\lceil y\rceil,y+r\}).$$ If $|\lceil y+r\rceil-1|$ is a non-zero perfect square and $\lceil y+r\rceil-1>\lfloor y\rfloor$, add $$(\lceil y+r\rceil-1,y+r).$$