Timeline for Strange real functions
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 15, 2010 at 5:40 | comment | added | fedja | Sure, Joel, if you tell which part of it gives you trouble (just to write essentially the same text in another window won't make much sense) | |
| Dec 12, 2010 at 12:23 | comment | added | Joel David Hamkins | Fedja, could you expand your comment into an answer? | |
| Dec 11, 2010 at 23:25 | comment | added | fedja | Ah, OK. Take the Weierstrass function $\sum_k (n_k)^{-1/2}\sin n_k x$ with very fast growing $n_k$ then. Each point outside an exceptional set of measure $0$ lies in a "valley" of each term with sufficiently large index (the previous terms have too small derivatives and the following terms are too small to change that). So, for almost every point $x$, the portion of $y$ with $f(y)>f(x)$ is noticeable on both sides of $x$ in a sequence of intervals centered at $x$. But a.e. point of a set of positive measure is a density point. | |
| Dec 11, 2010 at 20:44 | answer | added | Joel David Hamkins | timeline score: 5 | |
| Dec 11, 2010 at 19:26 | answer | added | Fedor Petrov | timeline score: 1 | |
| Dec 11, 2010 at 19:13 | answer | added | Stefan Geschke | timeline score: 4 | |
| Dec 11, 2010 at 18:07 | comment | added | Willie Wong | @fedja: Portland added the option that the set is not dense, but has positive measure. | |
| Dec 11, 2010 at 15:14 | comment | added | fedja | Not sure what the change was but Mariano's answer remains valid: $|x-c|$ is a counterexample where $c$ is any inner point of the domain. | |
| Dec 11, 2010 at 14:14 | history | edited | Portland | CC BY-SA 2.5 |
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| Dec 11, 2010 at 14:14 | comment | added | Portland | @Mariano, indeed it is obvious so I slightly changed the question. | |
| Dec 11, 2010 at 13:55 | comment | added | Joel David Hamkins | The first question is a duplicate of mathoverflow.net/questions/32126/…. | |
| Dec 11, 2010 at 13:50 | comment | added | Mariano Suárez-Álvarez | The restriction of the absolute value function to $[-1,1]$ does not seem to be monotonous on any dense subset of $[-1,1]$. | |
| Dec 11, 2010 at 13:49 | history | edited | Portland | CC BY-SA 2.5 |
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| Dec 11, 2010 at 13:43 | history | asked | Portland | CC BY-SA 2.5 |