We introduce the persistent homotopy type distance d HT to compare two real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition 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 homotopy equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that d HT still

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... s 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 L ∞ distance and the natural pseudo-distance d NP . From a different standpoint, we prove that d HT extends the L ∞ distance and d NP in two ways. First, we show that, appropriately restricting the category of objects to which d HT applies, it can be made to coincide with the other two distances. Finally, we show that d HT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.