Problem
Give an example of a function $\phi : \mathbb{R}^2 \to \mathbb{R}$ such that $$\phi(av) = a\phi(v)$$ for all $a \in \mathbb{R}$ and all $v \in \mathbb{R}^2$ but $\phi$ is not linear.
Solution
Let $\phi((x,y)) = \textrm{sgn}(x)\sqrt{x^2 + y^2}$, then: $$\phi(a(x,y)) = \phi((ax,ay)) = \textrm{sgn}(ax)|a|\sqrt{x^2 + y^2}$$ $$= a \cdot \textrm{sgn}(x)\sqrt{x^2 + y^2} = a \phi((x,y))$$ But consider $(0,1)$ and $(1,0)$, then: $$\sqrt{2} = \phi((1,1)) = \phi((1,0) + (0,1)) = \phi((1,0)) + \phi((0,1)) = 1 + 1 = 2$$ A contradiction.