JEE Math Practice Question

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements (S1) : $$A \cap B=(1, \infty)-\mathbb{N}$$ and (S2) : $$A \cup B=(1, \infty)$$

1. A.only $$(\mathrm{S} 2)$$ is true
2. B.only (S1) is true
3. C.neither (S1) nor (S2) is true
4. D.both (S1) and (S2) are true