<s>[INST] Is $d(x,y)=|x^3-y^3|$ metric on $\mathbb R$? What about property $2$:
$d(x,y)=0$ iff $x=y$
$|x^3-y^3|=0$
$x^3-y^3=0$
$(x-y)(x^2+xy+y^2)=0$
If $x-y=0$ then $x=y$. 
But if $x^2+xy+y^2=0$ then $x$ does not equal to $y$.
 [/INST] Note that  $$x^2+xy+y^2= (x+\frac {y}{2} )^2 + \frac {3}{4}y^2$$
Thus $$x^2+xy+y^2=0 \iff x=y=0$$
Therefore there is no problem with your metric.    
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