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]: https://math.stackexchange.com/questions/4347133/do-these-hyper-discontinuous-functions-exist
[2]: https://math.stackexchange.com/a/5025279/125918
#measure #continuity #discontinuity #function