Conditions for a map to be $\mathbb{R}$-linear Given a map $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, is the condition that for all $x,y$ in $\mathbb{R}^m$, $f(x+y)=f(x)+f(y)$ enough for $f$ to be $\mathbb{R}$-linear? I cannot prove it or think of a counter example.
It should also satisfy:
$$f(ax)=af(x).$$
It should also satisfy:
$$f(ax)=af(x).$$