Question 5
a) Prove that there exist non-periodical functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$
f(x + 1) + f(x - 1) = \sqrt{5}f(x),
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g : \mathbb{R} \to \mathbb{R}$ such that
$$
g(x + 1) + g(x - 1) = \sqrt{3}g(x),
$$
for all $x \in \mathbb{R}$, is periodical.
$$
f(x + 1) + f(x - 1) = \sqrt{5}f(x),
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g : \mathbb{R} \to \mathbb{R}$ such that
$$
g(x + 1) + g(x - 1) = \sqrt{3}g(x),
$$
for all $x \in \mathbb{R}$, is periodical.