Let $f : \mathbb{R} \to \mathbb{R}$ be an additive function, that is for all $x, y \in \mathbb{R}$, $f(x+y) = f(x) + f(y)$.
Prove $f(0)= 0$
I'm a little rusty on proofs. Can I just say suppose $x=0$ and $y=0$, $$f(x+y) = f(x) +f(y)$$
$$f(0+0) = f(0) + f(0)$$
$$f(0) = 2f(0)$$
$$f(0) = 0$$
I have no idea if that's the right process at all. Any hints or help would be greatly appreciated! Thank you