Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we say about f(x)

Question

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we deduce about $f(x)$?

I intend to say that $f(x)=x$, but find difficult to prove it. Is my guess correct, or wrong?

Answer 1

Let $y=-x$. THen

$$0=f(0)=f(x+-x)=f(x)+f(-x)$$

Then

$$f(x)=-f(-x)$$ or $$-f(x)=f(-x)$$

What do you know about equations that have this form?