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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 stilldoi:10.4310/hha.2019.v21.n2.a13 fatcat:qsxdthg2hjgnzfscdnvvccpz2m