Timeline for Does there exist a “fat” Thomae’s function?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2021 at 23:44 | comment | added | Nate River | @Dave L Renfro Nice! I would be curious to know about some of the unresolved problems you have. If you don’t mind, you could share some of them with me! | |
| Jun 19, 2021 at 19:02 | comment | added | Dave L Renfro | Update: A little over a week ago my contract work went from a nearly nonexistent trickle (past 2 months) to a wide open deluge, and there is also a huge amount of material I have on hand to sort through, so this will take a few more days. However, I think the result will be a useful survey of a lot of very little known literature, some of it I haven't thought about for 25+ years and some of it I've had unresolved questions about (from the 1990s) that I've never gotten around to sorting out (but I'm trying to now). | |
| Jun 17, 2021 at 2:03 | comment | added | Nate River | Ah it’s okay, take your time. | |
| Jun 16, 2021 at 17:10 | comment | added | Dave L Renfro | I'm not going to have the time today to write a summary of what I've found/know. I'll try to get something written tomorrow for an "answer". | |
| Jun 15, 2021 at 16:52 | comment | added | Dave L Renfro | and stronger than approximate continuity 1 2 (less clear). Moreover, if a function is everywhere measure continuous then it is everywhere continuous, which incidentally is not true for approximate continuity. However, in your case we don't want a function that is everywhere measure continuous. I believe you want both $C$ and $MC - C$ to be metrically dense sets. | |
| Jun 15, 2021 at 16:52 | comment | added | Dave L Renfro | I managed to look around a bit a few hours after my last comment and found some items and results that are more relevant, but I'm not going to have time to discuss them now and I also do not (yet) know the answer to your specific question. I'll try to write some kind of answer tomorrow, even if it only discusses what follows in a bit more detail. For now I'll mention that your generalized continuity condition is sometimes called "measure continuous", which unfortunately is a useless phrase for internet searching purposes. The notion is weaker than ordinary continuity (clear) (continued) | |
| Jun 15, 2021 at 14:50 | comment | added | Willie Wong | Comment on the terminology: typically one defines a discontinuity as essential on the real line if at least one of the one-sided limits fail to exist. The choice to call what you are defining an "essentially removable discontinuity" is a somewhat unfortunate clash. | |
| Jun 15, 2021 at 8:31 | comment | added | Dave L Renfro | Places to look (I don't have time now to dig into this): Henry Blumberg's papers (especially late 1910s through early 1920s, and measurable boundaries paper, and exceptional sets paper), Ákos Császár's early papers on negligent limits (e.g. 1 and 2 and 3 and 4), and Chapter 3 of Thomson's book. | |
| Jun 15, 2021 at 7:08 | history | asked | Nate River | CC BY-SA 4.0 |