已知符号函数 $\operatorname{sgn} x=\left\{\begin{array}{ll}1, x>0, & -1 \\ 0, & x=0, \\ -1, x < 0,\end{array}\right.$ 数 $f(x)$ 满足 $f(1-x)=f(1+x), f(x+2)=f(x)$, 当 $x \in[0,1]$ 时, $f(x)=\sin \left(\frac{\pi}{2} x\right)$, 则
$\text{A.}$ $\operatorname{sgn}(f(x))>0$
$\text{B.}$ $f\left(\frac{4041}{2}\right)=1$
$\text{C.}$ $\operatorname{sgn}(f(2 k))=0(k \in \mathbf{Z})$
$\text{D.}$ $\operatorname{sgn}(f(2 k))=|\operatorname{sgn} k|(k \in \mathbf{Z})$