ISL 2023 A2

Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$f(x+y) f(x-y) \geqslant f(x)^{2}-f(y)^{2}$$ for every $x, y \in \mathbb{R}$. Assume that the inequality is strict for some $x_{0}, y_{0} \in \mathbb{R}$. Prove that $f(x) \geqslant 0$ for every $x \in \mathbb{R}$ or $f(x) \leqslant 0$ for every $x \in \mathbb{R}$. (Malaysia)