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?
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 $\rho$.