Cauchy functional equation under certain conditions

Question

Does the equation $f(x+y)=f(x)+f(y)$ (with $x \in\mathbb{R^+}$ and $y \in\mathbb{R^-}$) ($0$ in included in $\mathbb{R^+}$ and $\mathbb{R^-}$) have the same solution of the normal equation knowing that we have that $f$ is bijective and increasing

Answer 1

Yes. For $x, y \in \mathbb R_+$,

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

Do the same for $x, y\in \mathbb R_-$ and you will have the "normal equation".