We consider the problem of interpolating functions partially defined over a distributive lattice by means of lattice polynomial functions. Goodstein's theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, respectively: given a function f : {0, 1} n → L , there exists a lattice polynomial function p:Ln→L such that p| {0,1} n = f if and only if f is monotone; in this case, the interpolating polynomial p is unique. We extend Goodstein’s theorem to a wider class of partial functions f:D→L over a distributive lattice L, not necessarily bounded, and where D⊆Ln is allowed to range over n-dimensional rectangular boxes D={a1,b1}×...×{an,bn} with ai,bi∈L and ai