Let $X\subseteq \mathbb{R}$ and $Y\subseteq\mathbb{R}$ be arbitrary sets, where we define a function $f:X\to Y$.
I need a measure of discontinuity which satisfies the requirments of this post. I want to see if my measure of discontinuity is infinite for all hyper-discontinuous functions?
Definition:
Suppose,
- $a,b\in\mathbb{R}$
- $G(a,b)$ is the graph of $\left.f\right|_{[a,b]}$
- $\mathbb{G}(a,b)$ is all limit points of $G(a,b)$
- $\ell\subset \mathbb{R}^2$ is a arbitrary non-vertical line, where $\left\{(x,y)\in\mathbb{R}^2: a< x < b, y=0\right\}\cap \ell \not=\emptyset$
- $|\cdot|$ is the absolute value
- $\#|\cdot|$ is the cardinality
- $\mathcal{D}$ is the measure of discontinuity, where
- $\mathcal{D}^{+}=\sup\limits_{a,b\in\mathbb{R}}\bigg(\limsup_\limits{|\text{slope}(\ell)|\to +\infty}\bigg\{z-1: z=\#\Big|\big(G(a,b)\cup\mathbb{G}(a,b)\big)\cap\ell\Big|\bigg\}\bigg)$
- $\small{\mathcal{D}^{-}=\inf\left(\left\{\sup\limits_{a,b\in\mathbb{R}}\bigg(\liminf\limits_{|\text{slope}(\ell)|\to +\infty}\bigg\{z-1: z=\#\Big|\big(G(a,b)\cup\mathbb{G}(a,b)\big)\cap\ell\Big|\bigg\}\bigg)\right\}\setminus\{-1\}\right)}$
- $\mathcal{D}=(\mathcal{D}^{+}+\mathcal{D}^{-})/2$
Example of Hyper-Discontinuous Function where $\mathcal{D}=+\infty$:
Consider $f:\mathbb{Q}\cap[0,1]\to\mathbb{Q}\cap[0,1]$, where $f(p/q)=1/q$.
Suppose, $\ell(x)=(2^n)x-(2^{n-1})$, where $n\in\mathbb{N}$ as $n\to\infty$, and $|\text{slope}(\ell)|\to+\infty$.
For every $n\in\mathbb{N}$, all interesections between $\ell$ and $f$ is:
$${I}=\left\{\left(\frac{2^n+\alpha(2^n-1)}{2^n+1+\alpha(2^n)},\frac{1}{2^n+1+\alpha(2^n)}\right):\alpha\in\mathbb{N}\cup\left\{0\right\}\right\}$$
Notice, for all $n\in\mathbb{N}$, the counting measure of $I$ is $+\infty$
Hence, $\mathcal{D}^{+}=+\infty$, making $\mathcal{D}=+\infty$
Recap of Question: Is there a hyper-discontinuous function where $\mathcal{D}<+\infty$?
