AoPS Wiki talk:Problem of the Day/September 19, 2011

We know that $f(f(x))=x$ for this particular function. Since $(x,y)$ is on the graph, $y=f(x)$ ; since $(y,z)$ is on the graph, $z=f(y)=f(f(x))=x$ . Thus, $f(x)=f(z)$ , so the answer is $\boxed{\textbf{(C)}}$ .