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 O 1 and O 2 of G, |f(O 1)-f(O 2)| ≤ 1 whenever O 1 and O 2 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 O 1 and O 2 of G with f(O 1) = p and f(O 2) = 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 arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.