smc

We can start by finding the value of $f(0)$. Let $a=b=0$ $$f(0)+f(0)=2f(0)+2f(0)\Rrightarrow 2f(0)=4f(0)\Rrightarrow f(0)=0$$ Thus, $A$ is not true. To check $B$, we let $a=0$. We have $$f(0+b)+f(0-b)=2f(0)+2f(b)$$ $$f(b)+f(-b)=0+2f(b)$$ $$f(-b)=f(b)$$ Thus, $B$ is not true, but $C$ is. Thus, the answer is $\boxed{f(-x)=f(x)}$