The conditions for $d:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ be a metric are:
d1) $d(x,x)=0$
d2) If $x\neq y$, $d(x,y)>0$
d3) $d(x,y)=d(y,x)$
d4) $d(x,z)\leq d(x,y)+d(y,z)$
I'm trying to find functions $d$ such that $d$ does not satisfies one of that conditions, but satisfies the another three.
$d_{1}(x,y)=0$ for all $x,y\in\mathbb{R}$ satisfies d1, d3 and d4, but not d2.
$d_{2}(x,y)=1$ for all $x,y\in\mathbb{R}$ satisfies d2, d3 and d4, but not d1.
Can someone give me some hints to find the other functions? Preferably I want just hints and ways to think, not the answers.