The theory of multiparameter persistent homology was initially developed in the discrete setting of filtered simplicial complexes. Stability of persistence was proved for topological spaces filtered by continuous vector-valued functions. Our aim is to provide a formal bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. The existence of this bridge is not obvious for two reasons. One is related to the Sard’s lemma, namely, we do not have the isolation of critical values for generic vector functions. The other one is due to the phenomenon of structural gap between the two settings which appears in the multi-parameter case when using the standard piecewise linear interpolation of the discrete model.