Timeline for Continuous functions taking uncountably many values countably often
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2019 at 13:00 | vote | accept | Ivan Meir | ||
| Nov 12, 2019 at 12:45 | comment | added | Ivan Meir | @MateuszKwaśnicki Ah thank you that's interesting I will take a look at those references, | |
| Nov 12, 2019 at 7:05 | comment | added | Mateusz Kwaśnicki | @IvanMeir: I think the Weierstrass function typically has finite or uncountable (Cantor-like) level sets. For the former, see here; for the latter, "dimension of Weierstrass function" is likely a good search term; see here. | |
| Nov 12, 2019 at 1:45 | comment | added | Ivan Meir | @MateuszKwaśnicki Thank you for this example. I had a thought when looking at your answer which is that the Weirstrass Function should also work in a similar way? | |
| Nov 12, 2019 at 1:31 | comment | added | Iosif Pinelis | I think this works. | |
| Nov 11, 2019 at 23:19 | comment | added | Mateusz Kwaśnicki | @IosifPinelis: I added a somewhat more rigorous argument. | |
| Nov 11, 2019 at 23:18 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Nov 11, 2019 at 22:57 | comment | added | Iosif Pinelis | Thank you for answering this question. Can you also detail the statement "intersects exactly one zigzag of $f$ of horizontal length $3^{-n}$"? | |
| Nov 11, 2019 at 22:51 | comment | added | Mateusz Kwaśnicki | @IosifPinelis: Yes, I did, thanks. Regarding the second question: $f(x) = C(x)$ outside of flat intervals of $C(x)$. If $y_0$ is not a dyadic rational, there is exactly one $x$ which $C(x) = y_0$, and this $x$ is not in any of the flat intervals. I'll try to clarify the answer momentarily. | |
| Nov 11, 2019 at 22:44 | comment | added | Iosif Pinelis | In place of $f(x) = C(x) = x$, did you mean $f(x) = C(x) = y_0$? If so, how will you make $f(x) = C(x)$ happen? | |
| Nov 11, 2019 at 22:32 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Nov 11, 2019 at 22:25 | comment | added | Mateusz Kwaśnicki | @StevenStadnicki: I do not think so, the $n$-th level bumps are only $2^{-n}$ high. | |
| Nov 11, 2019 at 22:22 | comment | added | Steven Stadnicki | This would seem to suffer the same continuity-based issues as the proposed solution in the comments... | |
| Nov 11, 2019 at 22:08 | history | undeleted | Mateusz Kwaśnicki | ||
| Nov 11, 2019 at 22:07 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Nov 11, 2019 at 21:45 | history | deleted | Mateusz Kwaśnicki | via Vote | |
| Nov 11, 2019 at 21:41 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |