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This is the question and its answer: enter image description here

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It is not clear for me why in the answer the author said about letter (B):"$\rho$ behaves like the Euclidean metric for values close together, and behaves like the discrete metric elsewhere ", could anyone clarify this for me please?

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  • $\begingroup$ If $x$ and $y$ are less than $1$ unit apart, then B just outputs their difference. That's Euclidean. If they are more than $1$ unit apart, it outputs $1$, which is what the discrete metric does $\endgroup$
    – The Chaz 2.0
    Jul 11, 2017 at 14:37

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If values are close together, then $\rho(x,y)=|x-y|$ and if they are apart then $\rho(x,y)=1$.

Thus, it behaves like Euclidean metric if the values are close(distance less than $1$) and like the discrete metric if the distace between the two points is greater than $1$.

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If $|x-y| < 1$ (i.e. $x$ and $y$ are close together) then $\rho(x,y) = |x-y|$

Otherwise they are far apart, an we don't care about the Euclidean distance anymore and just say $\rho(x,y) = 1$.

As all $\epsilon, \delta$ definitions are really about small values, effectively (for continuity/convergence etc.) it does not matter whether we use the Euclidean metric or $\rho$.

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  • $\begingroup$ what is the idea that you want to convey to me by saying this statement "As all ϵ,δ definitions are really about small values, effectively (for continuity/convergence etc.) it does not matter whether we use the Euclidean metric or ρ." $\endgroup$
    – user426277
    Jul 11, 2017 at 15:41
  • $\begingroup$ @Idonotknow that these metrics are equivalent. they both induce the standard topology. $\endgroup$
    – Henno Brandsma
    Jul 11, 2017 at 16:13

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