How to find a nonlinear function $f:mathbb{R}^2tomathbb{R}^2$ that is almost linear in the sense $f(alpha (a,b))=alpha f(a,b)$?

I need to find a nonlinear function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $f(\alpha (a,b))=\alpha f(a,b)$ for all $(a,b)\in\mathbb{R}^2$ and $\alpha\in\mathbb{R}$.

I can't find anything.

Context

The requirement $f(\alpha (a,b))=\alpha f(a,b)$ says that $f$ respects the scalar multiplication, just as linear maps do. In particular, $f$ is homogeneous of degree $1$. To make it nonlinear, one has to somehow destroy the additive property $f(a+c,b+d)=f(a,b)+f(c,d)$.