A question on distance functions and continuity in $\Bbb R^n$

Question

$d(x,y)=\|x-y\|$, $x,y \in \mathbb{R}^n$. How can I show $\|\cdot\|$ is continuous jointly in $x$ and $y$?

I have written the following:

$\|x-y\|<S$, $d(\|x-y\|,\|x_0-y_0\|)<\epsilon$ isn't that true? How should I continue?

Answer 1

$\forall \, x,y,x_0,y_0 \in \mathbb{R^n},$ we have

$$ |d(x,y)-d(x_0,y_0)|=|\, \|x-y\|-\|x_0-y_0 \| \,| \leq |\, ||(x-y)-(x_0-y_0)|| \,| $$

$$ = |\, ||(x-x_0)-(y-y_0)|| \,| \leq \| x-x_0\| + \| y-y_0\|< \delta_1 + \delta_2 =\epsilon,$$

where $\delta_1=\delta_2=\frac{\epsilon}{2}.$