# Continuity of the Shadow of a Nondecreasing Function

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.)

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Such functions appear in economics in jstor.org/stable/2297471?seq=7 I remember the author Novshek studying these functions in the context of a proof of the existence of equilibrium in certain games with monotone best responses. He established a fixed point theorem (whose proof to this day I don't understand) that seems to be interesting using these \emph{backward reaction functions}. They also appear in auction theory. The trick is always perturb the function so that your function is the identity (strict monotone functions) and ignore the ones with flats. –  Rabee Tourky Oct 19 '12 at 0:07
Do you mean $\: \mathcal{L}_f(x) = \operatorname{inf}(\{y\in [0,1] : f(x)\leq f(y)\}) \:$? $\;\;$ Your definition $\hspace{1.3 in}$ gives ugly answers when $f$ is discontinuous. $\;\;\;\;$ –  Ricky Demer Oct 19 '12 at 1:42
Ricky: your definition coincides with mine when the function $f$ is nondecreasing, if I'm not mistaken. I agree, my definition is ugly when $f$ is more general, but in that case you have to make some restriction on $f$ so that it's at least measurable, for instance, to make any sense of the fiber over a value. –  A Blumenthal Oct 19 '12 at 17:21
With $H$ as in en.wikipedia.org/wiki/Heaviside_step_function, our definitions disagree on $x\mapsto H\left(\frac12+x\right) \:$. $\;\;$ –  Ricky Demer Oct 19 '12 at 18:20