Choose a "middle" point from a set I read a post which talks about pretty much the same problem. But here I simplify the problem hoping that a concrete proof can be offered.

There is a set $A$ which contains some discrete points (one-dimensional), like $\{1, 3, 37, 59\}$. I want to pick one point from $A$ which minimizes the sum of distances between this point and others.

There may be lot of posts out there, and my problem is just the one-dimensional version of those. I know how to prove it if $A$ is not discrete, but I fail when $A$ is discrete like above.

Please answer with a concrete proof.
For a point $x$, let $d(x)$ be the sum of distances between $x$ and points in $A$. For $x \notin A$, the derivative $d'(x)$ has the nice formula
$$ d'(x) = |\{y \in A : y &lt; x\}| - |\{y \in A : y &gt; x\}|. $$
This shows why the median is the best answer when you don't have to select a point from $A$. For a point $x \in A$, $d(x)$ is the same as your objective function, hence the solution is to choose the median. You can find the median in linear time, as described in Wikipedia and various other resources.