Timeline for Continuity of the Shadow of a Nondecreasing Function
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2012 at 18:20 | comment | added | user5810 | With $H$ as in en.wikipedia.org/wiki/Heaviside_step_function, our definitions disagree on $x\mapsto H\left(\frac12+x\right) \:$. $\;\;$ | |
| Oct 19, 2012 at 17:21 | comment | added | A Blumenthal | 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. | |
| Oct 19, 2012 at 1:42 | comment | added | user5810 | 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. $\;\;\;\;$ | |
| Oct 19, 2012 at 0:07 | comment | added | Rabee Tourky | 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. | |
| Oct 18, 2012 at 22:36 | history | asked | A Blumenthal | CC BY-SA 3.0 |