$\text{ Period of }f\left(x\right)=\mathrm{sgn}\left(\right[x]+[-x\left]\right)\text{ is equal to (where [.] denotes greatest integer function) }$

 $\left[x\right]+\left[-x\right] =0 if x\in Z$

                      =$-1 if x\notin Z$ 

$Now f\left(x\right) =Sgn\left(\left[x\right]+\left[-x\right]\right) =Sgn\left(0\right) =0 if x\in Z$

                                            $= Sgn \left(-1\right) =-1 if x \notin Z$

$therefore f\left(x \right) period is 1$