Abstract. Consider a 1 , . . . , a n ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of a j -periodic functions, i.e., f = f 1 +· · ·+f n with ∆ a j f (x) := f (x+a j )−f (x) = 0. We show that f has such a decomposition if and only if for all partitions B 1 ∪B 2 ∪· · ·∪B N = {a 1 , . . . , a n } with B j consisting of commensurable elements with least common multiplesActually, we prove a more general result for periodic decompositions of functions f : A → R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f : A → R with respect to commuting, invertible self-mappings of some abstract set A.We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.