L^1's derivative is a perfectly good function, it's not defined or continuous at 0, but whatever... it's for the same reason that the median handles even-sized sets in a special way.

0 = d/ds \sum_i | x_i - s | = \sum_i sign(x_i - s)

which is true precisely when x_i <= s for half the elements and x_i >= s for the other half (taking sign(0) = 0, sign(+x) = 1, sign(-x) = -1).