continuousNotHolderContinuous

# Title: continuousNotHolderContinuous

## Overview

Designed to illustrate that Hölder continuity does not share a "bad" property of ordinary continuity, viz., being trivially satisfied in most model spaces of interest.

## Models

Our model spaces are subsets of real space with the standard metric.

Let $M \subseteq \mathbb R^2$ be the origin, which we label $C_M$, together with the unit circle centred on the origin, parameterised by $\theta$ running over the half-open interval $(0, 1]$.

Let $N \subseteq \mathbb R$ be $\{0\} \cup [1,+\infty)$.

## Consistency

We say that the origin $C_M$ is consistent only with $0$, while the point on the unit circle at parameter $\theta$ is consistent only with $1/\theta$.

## Consistency Restoration

This example is boring from the point of view of consistency restoration, as consistency is a bijective relation here so there is no choice about how to restore consistency, if we wish to be correct and hippocratic.

## Properties

Correct and hippocratic. All the properties that follow from that and the bijectiveness of the consistency relation, e.g., strongly undoable.

Continuous, but not Hölder continuous, see discussion.

## Discussion

A drawback of continuity as a property of bx is that it is trivially satisfied at any isolated point: we just pick $\delta$ small enough that there is no other point that close to the source and then the continuity condition holds vacuously. This is not true of Hölder continuity, leading to speculation that the latter might be a more useful property of bx, where model spaces are typically discrete.

Although $\overrightarrow{R}$ is continuous at every $(m,n)$, it is not $(C,\alpha)$-Hölder continuous at $(C_M, n)$ for any $n\in N$ because, while the distance between $C_M$ and any other $m'$ is 1, the distance between $\overrightarrow{R}(C_M, n) = 0$ and $\overrightarrow{R}(m',n)$ can be made arbitrarily large by judicious choice of $m'$.

## References

This is Example 4.8 from

@inproceedings{cheney15:bx,
author = {James Cheney and Jeremy Gibbons and James McKinna and Perdita Stevens},
title = {Towards a Principle of Least Surprise for Bidirectional Transformations},
booktitle = {Proceedings of {Bx} 2015},
year = 2015,
ee = {http://ceur-ws.org/Vol-1396/p66-cheney.pdf},
publisher = {CEUR-WS.org},
series = {CEUR Workshop Proceedings},
volume = 1396
}

Perdita Stevens