Prove that $f(x)=x+y^2+xy$ is continuous as a function $\mathbb{R}^2\rightarrow \mathbb{R}$ in the metric $d_1(x,y)=|x_1-y_1|+|x_2-y_2|$. (assume that on $\mathbb{R}$ we consider the usual absolute value metric).
I am at a loss as to how to prove them, because I am not quite sure on how to bound it such that it will be $<\epsilon$.
Let $0<\epsilon <1$. Fix $(x,y)$. Let us prove continuity at this point. $$|(x+y^{2}+xy)-(x'+y'^{2}+x'y')|$$ $$\leq |x-x'|+ |y-y'|(2|y|+|y-y'|)+|x||y-y'|+|y'||x-x'|$$. It is enough to make sure that $|x-x'|<\epsilon /4, |x-x'|<\epsilon /{4(|y|+1)},|y-y'| <\epsilon /(4|x|)$ and $|y-y'| <\epsilon/ {4(2|y|+1)}$