Derivations and differential operators on rings and fields

Let R be an integral domain of characteristic zero. We prove that a
function D : R → R is a derivation of order n if and only if D belongs to the
closure of the set of differential operators of degree n in the product topology
of R^R, where the image space is endowed with the discrete topology. In
other words, f is a derivation of order n if and only if, for every finite set
F ⊂ R, there is a differential operator D of degree n such that f = D
on F. We also prove that if d1, . . . , dn are nonzero derivations on R, then
d1 ◦ . . . ◦ dn is a derivation of exact order n.