Timeline for Is $δ=δ(x)$ a continuous function [closed]
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15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 24, 2020 at 10:54 | vote | accept | Safwane | ||
| Jan 13, 2020 at 20:25 | review | Reopen votes | |||
| Jan 14, 2020 at 12:25 | |||||
| Jan 13, 2020 at 19:52 | history | closed |
Alexandre Eremenko R W ARG user44191 LSpice |
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| Jan 13, 2020 at 19:45 | answer | added | Arno | timeline score: 5 | |
| Jan 12, 2020 at 14:20 | comment | added | Andrés E. Caicedo | And MR1837868 De Marco, Giuseppe For every $\epsilon$ there continuously exists a $\delta$.Amer. Math. Monthly 108 (2001), no. 5, 443–444. | |
| Jan 12, 2020 at 14:18 | comment | added | Andrés E. Caicedo | You may want to see also MR1745893 (2000m:54014) Enayat, Ali $\delta$ as a continuous function of $x$ and $\epsilon$. Amer. Math. Monthly 107 (2000), no. 2, 151–155. | |
| Jan 12, 2020 at 14:15 | review | Close votes | |||
| Jan 13, 2020 at 19:55 | |||||
| Jan 12, 2020 at 10:27 | comment | added | GH from MO | Then trivially the answer to your question is no. For any positive valued function, there is a smaller positive valued non-continuous function. See my first remark. | |
| Jan 12, 2020 at 10:25 | comment | added | Safwane | @GHfromMO: It can be considered as one of those reals which I assume depends on $x$ | |
| Jan 12, 2020 at 10:24 | comment | added | GH from MO | No, I meant that for a given $x$, infinitely many positive $\delta$'s work (if $\delta$ is ok than any smaller $\delta$ is also ok), so it is not clear what you mean by $\delta(x)$. | |
| Jan 12, 2020 at 10:23 | comment | added | GH from MO | Note also that on any compact interval $I\subset\mathbb{R}$ you can choose $\delta=\delta(I)$ to be constant, since $f'(x)$ is positive and uniformly continuous on $I$. | |
| Jan 12, 2020 at 10:22 | comment | added | Safwane | @GHfromMO: Did you mean that we must add some thing about the set of $x$. | |
| Jan 12, 2020 at 10:15 | comment | added | GH from MO | You have not defined $\delta$ uniquely as a function of $x$. For example, if $f(x)=x$, then one can take $\delta=1$ for $x\in\mathbb{Q}$ and $\delta=2$ for $x\not\in\mathbb{Q}$, which is not continuous (obviously). | |
| Jan 12, 2020 at 8:56 | answer | added | Fedor Petrov | timeline score: 9 | |
| Jan 12, 2020 at 8:31 | history | asked | Safwane | CC BY-SA 4.0 |