Continuous functions on countable compact ordered sets as sums of their increments

Every continuous function from a countable compact linearly ordered set A into a Banach space V (vanishing at the least element of A) admits a representation as a sum of a series of its increments (in the topology of uniform convergence). This series converges to no other sum under rearrangements of its terms. A uniqueness result to the problem of representation of a regulated real function on the unit interval as a sum of a continuous and a steplike function is derived.