be easily altered to make it work in the setting of plateaus.

In our setting motivated by dynamical systems we do not make any assumption that would provide the standard notion of critical points. A function $f$ can include ’*plateaus*’, i.e. pieces of constant value. As a consequence the local extrema, where the components of the sublevel-set filtration are born or die are not necessary unique.

Let $${\mathsf{diag}}\in{\mathsf{Per}}$$ be a persistence diagram composed out of a finite number of points $\{p_1,p_2,\dots,p_n\}$.

Where the first point $p_1$ corresponds to the component of infinite persistence, we have $$\label{finpts}
    {\mathsf{diag}}= \left\{p_1 = (p_1,\infty),\ p_2 = (p_2^b,p_2^d),\ \dots,\ p_n = (p_n^b,p_n^d)\right\},$$ where it holds that $n < \infty$, and $$p_i^b\neq p_j^b\text{ for }i\neq j\text{, and }p_j^d > p^d_{j+1}.$$ Due to the Elder Rule [@curry; @comptop] property of the persistence map it must hold that $$\label{p1less}
  p_1 < p_j^b,\text{ for all }j>1.$$

The diagram ${\mathsf{diag}}$ has its associated *barcode*, which we denote by $B({\mathsf{diag}})$, and which is composed out of the following intervals $$\label{barcode}
  I_1 = [p_1,\infty),\quad I_2 = [p_2^b,p_2^d),\ \quad\dots,\quad I_n = [p_n^b,p_n^d).$$ Observe that from the following inclusion on barcodes must hold $$I_j \subset I_1.$$

We say that a function $f(z,\cdot)$ realizes the 