This is probably a silly question with either a very simple example or a very easy explanation for why it's not possible. But anyway, I was goofing around with my friend and asked him what his favorite integer between 1737 and 1739 was, and he responded with $\pi$ (I know; we're weird).
But that got me thinking, "Is there a metric that we could construct on $\mathbb{R}$ where $\pi$ is actually between 1737 and 1739?" (even though it still wouldn't be an integer). I struggled to come up with any examples.
Moreover, how does one go about 'creating' metrics? Do we just propose candidates for metrics, and test whether they satisfy the necessary criteria? Is there any intuition that I should have on what is or isn't a metric? For reference, my background is a linear analysis course that I am currently taking in my first year of grad school.
First there is a question of what "between" would mean in metric spaces in general. And there the only thing that comes to my mind right away is that it's a situation where equality holds in the triangle inequality, i.e. one says $y$ is "between" $x$ and $z$ precisely if $$ d(x,y) + d(y,z) = d(x,z). \tag 1 $$ One trivial way to put $\pi$ between $1737$ and $1739$ is this: let $$ f(x) = \begin{cases} 1738 & \text{if }x=\pi, \\ \pi & \text{if } x=1738, \\ x & \text{otherwise,} \end{cases} $$ and then let $D(x,y) = |f(x)-f(y)|.$
Your subject line says "Is there a metric for which any $x$ is between any $y$ and $z$?".
Suppose we construe the question thus: Is it true that for any $x$ and $y$ and $z$ we can find such a metric? In that case, the answer is "yes": just proceed as above.
But the question could reasonably be understood as follows: Is there one metric such that for any $x,y,z,$ we have $x$ between $y$ and $z$ (i.e. which metric it is does not depend on $x,y,z$). In that case, the answer is "no". To see why, think about $(1)$ above, and ask whether $x$ is between $y$ and $z$ in $(1).$