If $f(x+y)-f(x-y)=4xy$, then is $f(0)=0$ or $f(0)=1$?

Question

As in the title. It may be very simple, but I'm having difficulty finding the proper substitution.

Answer 1

With $y=x=z/2$ you obtain $f(z)=z^2$

Answer 2

You don't even need to solve it, just note that if $f_1(x)$ is some solution, then so is $f_1(x) + c$ for any $c \in \Bbb R$. Thus we have proven $$ \text{There are solutions }\implies\text{ There are solutions with }f(0) \neq 0,1 $$ which is all we really need.