$\text{ The function }f\left(x\right)={\cos }^{-1}(\cos x)\text{ is }$

$f\left(x\right)={\cos }^{-1}\left(\cos x\right)=\left\{\begin{array}{l}2\pi +x\text{ for }-2\pi <x<-\pi \\ -x\text{ for }-\pi \le x<0\\ x\text{ for }0\le x\le \pi \\ 2\pi -x\text{ for }\pi <x<2\pi \end{array}\right.$

$f\left(x\right)\text{ is continuous at }x=\pi ,-\pi$