Let $\displaystyle \[f:\mathbb{R} \to \mathbb{R}\]$ such that:

$\displaystyle \[\tfrac{{\sqrt 5 + 1}}{2}f\left( x \right) = f\left( {x + 1} \right) + f\left( {x - 1} \right)\]$ (1)

a) Prove that f is periodic

b) Prove that there are infinitely many functions such that (1) is true