Timeline for What is the smallest set of real continuous functions generating all rational numbers by iteration?
Current License: CC BY-SA 4.0
12 events
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| Jan 30, 2022 at 12:13 | comment | added | Ivan Meir | That is surprising but quite satisfying! Saul thanks again for your efforts on this. | |
| Jan 29, 2022 at 8:40 | comment | added | Saúl RM | Surprisingly the function can even be taken to be analytic, here is a reference.Sorry for the wait, I thought a bit about the comment, forgot about it and recalled it today. | |
| Jan 21, 2022 at 0:54 | comment | added | Ivan Meir | That is very cool! So it looks like we can strengthen your result to Lipschitz continuous rather than simply continuous? This means absolutely continuous and therefore a.e. differentiable I believe. Is it correct that your function is not differentiable everywhere on $\mathbb{R}$? | |
| Jan 20, 2022 at 14:38 | comment | added | Saúl RM | Thanks! The homeomorphism from the second part of the answer can be Lipschitz, in fact given $A,B$ countable dense subsets of $\mathbb{R}$ and $\varepsilon>0$ there is an homeomorphism $h:\mathbb{R}\to\mathbb{R}$ with $h(A)=B$ and $(1-\varepsilon) d(x,y)<d(h(x),h(y))<(1+\varepsilon)d(x,y)\;\forall x,y$, as shown in theorem 13 of this paper. | |
| Jan 20, 2022 at 9:15 | comment | added | Ivan Meir | Thanks Saúl this is a lovely solution. I noted from the MO answer referenced below that you can generate the dense sequence using a Lipschitz continuous function. I am I right to assume that the homeomorphism you use for the second part of your answer is not Lipschitz? | |
| Jan 6, 2022 at 11:36 | vote | accept | Ivan Meir | ||
| Jan 3, 2022 at 10:26 | history | edited | Saúl RM | CC BY-SA 4.0 |
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| Jan 1, 2022 at 21:50 | history | edited | Saúl RM | CC BY-SA 4.0 |
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| Jan 1, 2022 at 15:35 | history | edited | Joe Silverman | CC BY-SA 4.0 |
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| Jan 1, 2022 at 15:11 | history | edited | Saúl RM | CC BY-SA 4.0 |
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| Jan 1, 2022 at 14:36 | history | edited | Saúl RM | CC BY-SA 4.0 |
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| Jan 1, 2022 at 13:24 | history | answered | Saúl RM | CC BY-SA 4.0 |