Which of the following functions are periodic? 

$A) f\left(x\right)=\mathrm{sgn}\left(e^{-x}\right)$

  $e^{-x}>0$  for every real ‘x’
Every constant function is a periodic function 
B)  Let T > 0 be a rational number then f(x + T) = $\{\begin{array}{cc}1& x\in Q\\ 0& x\notin Q\end{array}=f\left(x\right)$

$\therefore$f(x) is periodic function

C)  $f\left(x\right)=\sqrt{\frac{8}{1-\cos x}+\frac{8}{1-\cos x}}=\sqrt{\frac{16}{1-{\cos }^{2}x}}=\frac{4}{\left|\sin x\right|}=4\left|\cos ecx\right|$

This is a periodic function 

D)  $f\left(x\right)=[x+\frac{1}{2}]+[x-\frac{1}{2}]+2[-x]$

      $=(x+\frac{1}{2}-\{x+\frac{1}{2}\}+x-\frac{1}{2}-\{x-\frac{1}{2}\})+2$

$+2(-x-\{-x\})=-\{x+\frac{1}{2}\}-\{x-\frac{1}{2}\}-2\{-x\}$

Since {.} is a periodic function hence this function is periodic.