Obtaining the usual metric on $\mathbb R$ by "pulling back" $\arctan$ This is a part from an e-mail that a professor wrote to me. I couldn't understand what he meant by

Consider two different metrics on the real line $\mathbb R$ inducing
  its usual topology: one is the usual metric, the other is the metric
  obtained by choosing a homeomorphism $\mathbb R \cong (0,1)$ (use
  arctan, for example) and “pulling back" the usual metric on $(0,1)$.

How did he define the second metric on $\mathbb R$? I guess it is something like
$$d(x,y) = \frac{2}{\pi} \arctan |x-y|$$ But of course my guess is not true. Can you give a formula for the "pullback" metric?
Thanks.

Define $f(x)=\frac{1}{\pi}\arctan x+\frac{1}{2}$.  
Then taking $d(x,y)=|f(x)-f(y)|$ gives $d(x,y)=\frac{1}{\pi}\left|\arctan x-\arctan y\right|$.
