I see there are some counterexamples and so forth in metric spaces regarding the metric $$d(x,y) = | \arctan(x) – \arctan(y)| $$
But honestly I have no intuition as to how it works
For example, in the Euclidean metric the intuition here is just length between two points and you have a good visual/physical intuition as to how it works.
Discrete metric fixes that length between two points to be 1 and can be visualized as a unit line segment, or an equilateral triangle.
But what is the intuition behind this metric?
$$d(x,y) = | \arctan(x) – \arctan(y)| $$
What is the metric space equipped with $d(x,y)$ referred to? How do people came up with it and is it possible to have some mental picture of how the metric works. Finally, what is so special about this metric?
Let me know if this is just a metric created for sake of counterexample.
Best Answer
Here is a quick graph and how $d(x,y)$ works. Notice that near $0$, $d(x,y)$ evaluates the distance of $x,y$ as if it were the absolute value, whereas for large $|x|$ things are completely different: distances are quite small. In fact the "biggest" distance you'd get is $\pi$ (and is never attained of course).