Timeline for On the existence of a sequence of positive continuous functions
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2010 at 5:34 | comment | added | Gerry Myerson | I'll leave up my last comment, even though Koel has edited out the motivation for it, since it may help anyone else who found the definition of $f_n$ in my answer opaque. | |
| Aug 31, 2010 at 5:05 | comment | added | Gerry Myerson | Let me illustrate with $f_3$. $f_3(x)=3$ for $x=0,1/3,1/2,2/3,1$. $f_3(x)=0$ for $x=1/81,1/3±1/81,1/2±1/81,2/3±1/81,1−1/81$. For all other $x$, $0\le x\le1$, connect the dots. So, e.g., $f_3(x)=0$ for all $x$ with $1/81\le x\le1/3−1/81$, whether such $x$ are rational or irrational. Strictly speaking, this only defines $f_3$ on $[0,1]$, but extend it to all of $\bf R$ by making it periodic with period one. | |
| Aug 30, 2010 at 7:52 | comment | added | user8840 | @ Henriksen The functions attain the value 1 on a rational for all but finitely many 'n' . Hence these would not give us continuous functions taking (only) rationals to infinity. | |
| Aug 28, 2010 at 7:01 | history | answered | K. Henriksen | CC BY-SA 2.5 |