Finding period of $f$ from the functional equation $f(x)+f(x+4)=f(x+2)+f(x+6)$ How can I find the period of real valued function satisfying $f(x)+f(x+4)=f(x+2)+f(x+6)$?
Note: Use of recurrence relations not allowed. Use of elementary algebraic manipulations is better!

Observe, since 
$$
f(x)+f(x+4)=f(x+2)+f(x+6),
$$
we can substitute $x+2$ for $x$ to get 
$$
f(x+2)+f(x+6)=f(x+4)+f(x+8).
$$
Equating these, we know that $f(x)=f(x+8)$.
