So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq f(y)$.
The dynamics is highly sensitive to the information $f(x) \leq f(y)$, and so I wanted to find a way to encode this. I came up with \begin{align*} \mathcal{L}_f(x) = \inf f^{-1}(f(x)) \end{align*} That is, check the fiber over a value of $f(x)$ and spit out its infimum. That $f$ is nondecreasing implies each fiber is either an interval (of some kind, depending on the discontinuities of $f$, but always of length $> 0$) or a single point. Likewise you can take the $\sup$ and come up with a function $\mathcal{R}_f$ that encodes the 'rightmost point' of each fiber. There are several equivalent ways to write such an object, but the one above seems the most intuitive.
Let me preface my questions with the following observation: it is painfully clear that the assignment $f \rightarrow \mathcal{L}_f$ is discontinuous in many of my favorite topologies on either the range or domain.
Question 1: What are $\mathcal{L}, \mathcal{R}$ called in the literature? I was told that these are called the 'shadows' of $f$, but I can't find any reference to such nomenclature. Is there some book on analysis which defines/names these objects? The idea seems elementary enough (the definition evokes the Skorohod construction, for e.g.).
Question 2: Are there any (interesting / nontrivial) topologies on the space of nondecreasing functions for which the assignment $f \rightarrow \mathcal{L}_f$ is continuous? In other words, are there any modes of convergence $f_n \rightarrow f$ which will ensure $\mathcal{L}_{f_n} \rightarrow \mathcal{L}_{f}$ in some sense, perhaps weakly? (Caveat: none of uniform, $L^p$, vague, pointwise a.e. will work.)
