The domain of the function f defined by

$f(x)=f(x)=\frac{1}{\sqrt{x-|x|}}$

f(x) is defined if  x − |x| > 0

i.e  x > |x| 

i.e |x| < x

Since no such real number x exist such that |x| < x

$\therefore$ Domain of $f(x)$ is empty set.

Hence, the correct answer is option D.