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#continuity

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Bharath Krishnan

Motivation:

Let X⊆R and Y⊆R be arbitrary sets, where we define a function f:X→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 +∞
• When f is "hyper-discontinuous" [1], the measure is +∞
• When the graph of f is dense in the derived set of X×Y, the measure is +∞
• 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...
#measure#continuity#discontinuity
Public
Wen

Continuity/Fact checking of the moment - Prime Target

Apple have been showing ‘Prime Target’ - an occasionally amusing, almost always silly thriller based around the premise that codes based on prime numbers can be easily cracked (short form explanation). But when our heroes make of in a #Swastikar (Tesla) the dubbed sound is of a combustion engine 🙄. Who got that wrong?

Well it amused, even if the history and maths were suspect

#PrimeTarget#Silly#AppleTV
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Rainer AI Blockchain Rehak 4.0

“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
#deportation#Gaza#continuity
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Markus Redeker

Somewhere I just read that that category-theoretic continuity is similar to topological continuity because in both cases two operations can be interchanged: With categories, a functor commutes with a limes construction, while in topology, we have for example

limn→∞f(an)=f(limn→∞an).

But what is really going on is that both concepts are so useful because they introduce, in a special context, commutativity. Whenever you can exchange operations, you can simplify formulas and life becomes easier.

This is the inner-mathematical reason why continuity (in both senses) is so useful.

#Mathematics#Topology#CategoryTheory
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