I was reading a text that seemed to say:
If $nf(x)=f(nx)$ for all n in $\mathbb{N}$, then $f(x)$ is proportional to $x$ (or $f(x)=kx$ for $k$ in $\mathbb{R}$).
I feel like this is probably the case, but I can't think of a good reason why it has to be.
I can't either. Let $f(x)=|x|$.