Timeline for On the definition of a continuous function
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 24, 2022 at 3:17 | comment | added | mamediz | This graph is connected but it is not locally connected, anyway, this example shows that it would be interesting to change the definition to require that the graph must be locally connected at (a,f(a)) instead of just connected for f restricted to some neighborhood of a. | |
| Oct 23, 2022 at 7:30 | comment | added | KP Hart | @MadeleineBirchfield probably not, given Brouwer's theorem that fully defined functions are continuous. | |
| Oct 23, 2022 at 0:53 | comment | added | Madeleine Birchfield | Are there any concrete examples in constructive real analysis? In constructive mathematics, the given function in this answer can't be proven to be path-continuous because it is only defined on $x = 0$, $x > 0$ and $x < 0$. Absent excluded middle, this is not equivalent to being defined on the entire set of real numbers because trichotomy can't be proven. | |
| Oct 22, 2022 at 23:25 | comment | added | Wlod AA | @PiotrHajlasz, I said in my mind. (Right, of course, u'r not a reader of my mind, and let's keep it this way). | |
| Oct 22, 2022 at 22:58 | comment | added | D.S. Lipham | What you describe is essentially Sierpinski's "punctiform connected graph" where punctiform = no compact connected subset. The OP may want to see the dissertation here: apronus.com/static/MRWojcikPhD.pdf, which is a fairy complete study of this type of thing. | |
| Oct 22, 2022 at 16:02 | comment | added | Piotr Hajlasz | @WlodAA I do not see any examples in your comments. You just said it is fale when $A=B=N_a=\mathbb{R}$. That is not an example. | |
| Oct 21, 2022 at 22:14 | comment | added | Saúl RM | There are functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(U)=\mathbb{R}$ for every nonempty open set $U$, so that's also a pretty discontinous counterexample | |
| Oct 21, 2022 at 10:54 | history | edited | KP Hart | CC BY-SA 4.0 |
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| Oct 21, 2022 at 9:06 | comment | added | Wlod AA | That was about the simplest of examples that I had in my mind when commenting on OP post. | |
| Oct 21, 2022 at 8:22 | history | answered | KP Hart | CC BY-SA 4.0 |