6. If $f(x)$ is a real-valued function of a real variable $x$, and $f(x)$ is not identically zero, and for all $a, b, f(a+b)+$ $f(a-b)=2 f(a)+2 f(b)$, then for all $x, y$
A. $f(0)=1$
B. $f(-x)=-f(x)$
C. $f(-x)=f(x)$
D. $f(x+y)=f(x)+f(y)$
E. There is a positive number $T$ such that $f(x+T)=f(x)$
6. C Let $a=b=0, f(0)=0$. Then let $a=0, b=-x$, we have $f(x)+f(-x)=2 f(-x), f(x)=$
$$
2 f(-x)-f(-x)=f(-x) .
$$