In algebraic topology it is well known that, using the Mayer–Vietoris sequence, the homology of a space X can be studied by splitting X into subspaces A and B and computing the homology of A , B , and A ∩ B . A natural question is: To what extent does persistent homology benefit from a similar property? In this paper we show that persistent homology has a Mayer–Vietoris sequence that is generally not exact but only of order 2. However, we obtain a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X , A , B , and A ∩ B plus three extra terms. This implies that persistent homological features of A and B can be found either as persistent homological features of X or of A ∩ B . As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.