The real valued function $$f(x) = \frac{\text{cosec}^{-1} x}{\sqrt{x - [x]}}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is defined for all $$x$$ belonging to:

The function is $$f(x) = \frac{\text{cosec}^{-1} x}{\sqrt{x - [x]}}$$, where $$[x]$$ is the greatest integer function. We need both the numerator and denominator to be defined, with the denominator nonzero.

For $$\text{cosec}^{-1} x$$ to be defined, we need $$|x| \geq 1$$, i.e., $$x \leq -1$$ or $$x \geq 1$$.

The denominator is $$\sqrt{x - [x]} = \sqrt{\{x\}}$$ where $$\{x\}$$ is the fractional part. For this to be defined and nonzero, we need $$\{x\} > 0$$, which means $$x$$ must not be an integer.

Combining both conditions: $$x$$ must satisfy $$|x| \geq 1$$ and $$x$$ must not be an integer. In other words, $$x$$ belongs to all non-integers outside the open interval $$(-1, 1)$$. Since at $$x = 1$$ and $$x = -1$$ (which are integers), the fractional part is zero, these points are excluded too.

Therefore the domain is all non-integers except those in $$[-1, 1]$$, which is equivalently stated as all non-integers except the interval $$[-1, 1]$$.

The answer is Option B: all non-integers except the interval $$[-1, 1]$$.