If , then F(x) is: (where {.} denotes fractional part function and [..] denotes greatest integer function and sgn (x) is a signum function)

(A) periodic with fundamental period 1
(B) even
(C) range is singleton
(D) identical to sgn 

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{sin\pi }\left[\mathrm{x}\right]}{\left\{\mathrm{x}\right\}} \\ \mathrm{As} \mathrm{fractional} \mathrm{part} \mathrm{is} 0 \mathrm{for} \mathrm{x}\in \mathrm{I}, \mathrm{hence} \mathrm{domain} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{R}-\mathrm{I} \\ \mathrm{Now} \left[\mathrm{x}\right] \mathrm{always} \mathrm{gives} \mathrm{integer} \mathrm{and} \mathrm{at} \mathrm{integral} \mathrm{multiple} \mathrm{of} \mathrm{\pi }, \mathrm{sine} \mathrm{is} 0, \mathrm{hence} \\ \mathrm{F}\left(\mathrm{x}\right)=0 \mathrm{for} \mathrm{x}\in \mathrm{R}-\mathrm{I} \\ \mathrm{Hence} \mathrm{range} \mathrm{of} \mathrm{F}\left(\mathrm{x}\right) \mathrm{is} \left\{0\right\}-\mathrm{a} \mathrm{singleton} \mathrm{set}...\mathrm{Option}\left(\mathrm{C}\right) \end{gathered}$$

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