Discontinuous additive function

Question

I was trying to prove the following proposition. Let $l:\mathbb{R}^{2}\to \mathbb{R}_{\geq 0} $ be a function s.t. $$ \begin{align*} 1. & l\left(x\right)=0 \iff x=0 \\ 2. & l\left(x\right) =l\left(-x\right)\\ 3. & l\left(x+y\right) \leq l\left(x\right)+l\left(y\right)\\ 4. & l\left(x+y\right)=l\left(x\right)+l\left(y\right)\iff x=\lambda y \end{align*} $$ For a $0<\lambda$. Then $l$ is the Euclidean norm. I now believe this to be false. A function $$f:\mathbb{R}^{2}\to\mathbb{R}^{2}$$ that is a discontinuous additive bijection s.t. for every $x,y\in \mathbb{R}^{2}$ the condition $$\exists \lambda >0 , f\left(x\right)=\lambda f\left(y\right)\iff \exists \delta >0 , x=\delta y$$ holds would be a counter example. Does a function of this sort exist?