\mathbb{E},$$ For any diagram ${\mathsf{diag}}\in{\mathsf{Per}}$, and for each point $(b,d)\in{\mathsf{diag}}$, it holds that $b\leq d$. Moreover, in the one-dimensional setting there is necessarily one point having the coordinates $(b',\infty)$.

Here we do not study in detail the structure and decomposition of the persistence map. We define the *persistence map*, as the map which takes a function $f\colon{\mathbb{R}}^N\times{{\mathcal{S}_N}}\to{\mathbb{R}}$ and a vector $z\in{\mathbb{R}}^N$ to the persistence diagram encoding $0$-th order persistence homology of ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$. Also, note that in our setting we deal only with the persistence map of degree $0$, associated with connected components.

It is known that the persistence map is $1$-Lipschitz, after equipping the category of persistent vector spaces with the *interleaving distance* [@chazal].

There exists a vast literature on algorithms computing the persistence diagram for general simplicial inputs, see e.g. [@comptop; @ZC] and references cited there. But, in the literature regarding computational homology the space of admissible functions is chosen in a way admitting a notion of *discrete Morse functions*, which in turn provides the standard notion of critical points. As a consequence, the cells where components of the sub-level set filtration are born/die can be indicated uniquely by algorithms computing the persistent homology. However, the algorithm can 