We study the finite-dimensional spaces V, that are invariant under the action of the finite differences operator . Concretely, we prove that if V is such an space, there exists a finite-dimensional translation invariant space W such that V ⊆ W. In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V = P ⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by Montel [7], which states that, if f: ℝ → ℂ is a continuous function satisfying for all t ∈ ℝ and certain h 1, h 2 ∈ ℝ∖{0} such that h 1/h 2 ∉ ℚ, then f(t) = a 0 + a 1 t + … +a m−1 t m−1 for all t ∈ ℝ and certain complex numbers a 0, a 1,…, a m−1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f(t) but also for complex valued distributions. Finally, we also consider the subspaces V that are Δ h 1 h 2…h m -invariant for all h 1,…, h m ∈ ℝ.