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For $x=(x_1, x_2, \ldots, x_n), y=(y_1, y_2, \ldots, y_n) \in \mathbb{R}^n.$ Define $d: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ as $$d(x,y)=|x_1-y_1| + |x_2-y_2| + \cdots + |x_n-y_n|.$$ Show that $d$ is continuous by using definition in Topology, that is $d$ is continuous if for every open subset $V$ of $\mathbb{R},$ the set $d^{-1}(V)$ is open in $\mathbb{R}^n \times \mathbb{R}^n$.

I know that open sets in $\mathbb{R}$ are just intervals, but I don't know what its inverse image under $d$ looks like, so I'm stuck. Is it possible to show continuity using this definition, or should I use the $\epsilon-\delta$ definition?

Thank you!

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    $\begingroup$ Open sets in $\mathbb{R}$ are not just intervals, but arbitrary unions of intervals. Since $d^{-1}(\bigcup_iU_i)=\bigcup_id^{-1}(U_i)$, then it is enough to only check the inverse image of intervals. $\endgroup$
    – logarithm
    May 16, 2019 at 10:35
  • $\begingroup$ What are the inverse images of intervals? Say if $z \in (a,b)$ for some interval $(a,b)$, it is not clear to me how to get $d^{-1}(z).$ Please help $\endgroup$
    – Mashed Potato
    May 16, 2019 at 10:44
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    $\begingroup$ FYI, this is the taxicab metric $\endgroup$
    – YuiTo Cheng
    May 16, 2019 at 10:52
  • $\begingroup$ One simpler approach is to prove that the taxicab metric and the Euclidean metric are equivalent. Then use math.stackexchange.com/questions/287285/… $\endgroup$
    – YuiTo Cheng
    May 16, 2019 at 11:13
  • $\begingroup$ If $d_1,...,d_n$ are continuous functions from a domain $D$ into $\Bbb R$ then $d(v)=\sum_{i=1}d_i(v)$ is continuous. So show that each $d_i((x,y))=|x_i-y_i|$ is continuous. $\endgroup$
    – DanielWainfleet
    May 16, 2019 at 12:15

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