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#continuity

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Motivation:

Let \(X\subseteq \mathbb{R}\) and \(Y\subseteq\mathbb{R}\) be arbitrary sets, where we define a function \(f:X\to Y\).

I want a measure of discontinuity which ranges from zero to positive infinity, where

• When the limit points of the graph of \(f\) is continuous almost everywhere, the measure is zero
• When the limit points of graph of \(f\) can be split into functions, where \(n\) of those functions are continuous almost everywhere, the measure is \(n-1\)
• When \(f\) is discrete, the measure is \(+\infty\)
• When \(f\) is "hyper-discontinuous" [1], the measure is \(+\infty\)
• When the graph of \(f\) is dense in the derived set of \(X\times Y\), the measure is \(+\infty\)
• When the measure of discontinuity is between zero and infinity, the more "disconnected" the graph of \(f\), the higher the measure of discontinuity.

Question 1: How do we fix the criteria in the motivation, so they are consistent with eachother?

Question 2: Is there a measure of discontinuity which gives what I want?

Attempt: I tried to answer [2] this using a previous question, but according to users it's needlessly complicated and likely is incorrect. I'm also struggling to explain why this answer [2] has potential.

[1]: math.stackexchange.com/questio

[2]: math.stackexchange.com/a/50252

Mathematics Stack ExchangeDo these "hyper-discontinuous" functions exist?Suppose $\ X\subset \mathbb{R}\ $ and $\ Y\subset \mathbb{R}.$ Definition: A function $\ f: X \to Y\ $ is hyper-discontinuous if for every $\ x\in X,\ \ \exists\ \delta>0,\ \varepsilon>0\ $ s...

“In November 1989, while #Netanyahu served as a junior minister in Shamir’s government, he spoke at Bar-Ilan University, the academic mecca of religious nationalism in Israel. He said:

#Israel should have taken advantage of the suppression of the demonstrations in #China in #Tiananmen Square, when the world’s attention was focused on what was happening in that country, to carry out #mass #expulsions among the #Arabs of the territories.“

therealnews.com/mblumenthal102

The Real News Network · Netanyahu’s Greater Israel Based on Expulsion and Annexation – Max Blumenthal on Reality Asserts Itself pt3By Max Blumenthal