Question 15
a) Prove that there exist non-periodic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equality
$$
f(x+1)+f(x-1)=\sqrt{5} f(x)
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies the equality
$$
g(x+1)+g(x-1)=\sqrt{3} g(x)
$$
for all $x \in \mathbb{R}$, is periodic.
$$
f(x+1)+f(x-1)=\sqrt{5} f(x)
$$
for all $x \in \mathbb{R}$.
b) Prove that any function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies the equality
$$
g(x+1)+g(x-1)=\sqrt{3} g(x)
$$
for all $x \in \mathbb{R}$, is periodic.