existence of function in R2 with some property

I would like to know if there exists a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ satisfying :

property 1 : $f$ is bounded

property 2 : $f$ is $1$-Lipshitz

property 3 : $|f(a)-f(b)|\geq C \min(1,|a-b|)$ for a suitable fixed constant $C>0$ (and for any $(a,b)\in \mathbb{R}^2\times \mathbb{R}^2$).

Thanks