Which of the following define a metric on $\mathbb{R}$?
$d_1(x,y) = \frac{|x|-|y|} {1+|x||y|}$
$d_2(x,y) = \sqrt{|x-y|}$
$d_3(x,y) = |f(x)-g(x)|$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is strictly monotonic increasing function.
Here is my attempt:
$d_1(x,y)$ satisfies all the three conditions.
$d_2(x,y)$ may fail to satisfy triangle inequality.
$d_3(x,y)$ is not a well defined function.
I am not sure whether i am correct or not? I need a proper justifications.
Thanks for giving me time.