Write the domain of the real function

Question:

Write the domain of the real function $f(x)=\sqrt{[x]-x}$.

Solution:

$[x]$ is the greatest integer function.

$[x] \leq x, \forall x \in R$

$\Rightarrow[x]-x \leq 0, \forall x \in R$

$\Rightarrow \sqrt{[x]-x}$ does not exist for any $x \in R$.

Domain $=\phi$

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