I'll present the persistent homotopy type distance d_HT to compare two real valued functions defi ned on possibly different but homotopy equivalent topological spaces. The underlying idea in the defi nition of d_HT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the supremum distance and by the natural pseudo-distance d_NP. From a different standpoint, d_HT extends the supremum distance and d_NP in two ways. First, appropriately restricting the category of objects to which d_HT applies, it can be made to coincide with the other two distances. Secondly, dHT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.