If you consider a distribution that has many medians:
$P(X=x) = \{(1, 0.25)(2,0.25),(3,0.25),(4,0.25)\} ,$ we know that this distribution has multiple medians between $2$ and $3$ if we define a median as a number where $P(X \geq c) \geq1/2$ and $P(X \leq c) \geq 1/2$. I basically want to prove that the set of all medians is the only set of numbers that minimizes $E[|X-a|]$.
Attempt: Distributions can have multiple medians, and these multiple medians are the only numbers that minimize the absolute value of the distance between the mean. That is, there is a 1-1 correspondence between the set of numbers that minimizes the absolute value of the distance between the mean and the set of medians. I am stuck in that I can't show a rigorous proof for this question.