Let $f : \mathbb{R}\longrightarrow\mathbb{R}$ be a function such that $f(x + y) + f(x − y) = f(xy)$ for all $x, y \in\mathbb{R}$. Is $f$ constant?!
I do get that the function is an even function but I don't where to go forward from that? Can $f$ be zero btw ?
With $y=0$, we see that $$2f(x)=f(0),$$ hence $f$ is constant. In particular, with $x=0$, we have $2f(0)=f(0)$, hence the constant must be $0$.