Let the sets $A$ and $B$ denote the domain and range respectively of the function $f\left(x\right)=\frac{1}{\sqrt{\left[x\right]-x}},$ where $\left[x\right]$ denotes the smallest integer greater than or equal to $x$. Then among the statements

$\left(S1\right): A\cap B=\left(1, \infty \right)-\mathrm{\mathbb{N}}$ and

$\left(S2\right): A\cup B=\left(1, \infty \right)$

• A

  Only $\left(S2\right)$ is true
• B

  Only $\left(S1\right)$ is true
• Correct answer
  C

  Neither $\left(S1\right)$ nor $\left(S2\right)$ is true
• D

  Both $\left(S1\right)$ and $\left(S2\right)$ are true

Hence, the correct answer is option: 'C'.