Interpolation theorem for a continuous function on orientations of a simple graph

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O₁ and O₂ of G, |f(O₁) - f(O₂)| ≤ 1 whenever O₁ and O₂ differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O₁ and O₂ of G with f(O₁) = p and f(O₂) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arcconnectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.