Define a metric $d:\Bbb R \times \Bbb R \to \Bbb R$ by $d(x,y)=|\tan^{-1}x-\tan^{-1}y|$. I want to show that $\Bbb R$ is NOT a Normed linear space with respect to the norm induced by the metric $d$.

To show this this I'm trying to show that at least one of the properties :

Transnational invariant: $d(x+a,y+a)=d(x,y)$ ,

Absolute homogeneity: $d(ax,ay)=|a|.d(x,y)$

does not hold. But I'm unable to find this.

Can anyone give some hint for this ?


Hint: Compute $d(\sqrt 3,0)$ and $d(\sqrt3/3,0)$.


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