Timeline for A question on a special type of function
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2011 at 9:57 | vote | accept | Jose Arnaldo Bebita | ||
| Mar 3, 2011 at 5:02 | comment | added | Joe | Do you want the first inequality to hold for all naturals x and y, or only when x is a square? Yes, the same construction will work. If we're defining $f(y)$, then for all sufficiently large choices of $f(y)$, $f(\sqrt{x})/y$ will stay small while $f(y)/\sqrt{x}$ becomes large. When this happens, the inequality will be greater than 1/2. The same happens if we're defining f at a square, for which we must consider it in the role of $x$ as well. | |
| Mar 2, 2011 at 9:58 | comment | added | Jose Arnaldo Bebita | @Joe, will this same construction work if the inequality $|\frac{f(\sqrt{x})}{y} - \frac{f(y)}{\sqrt{x}}| > 1$ is replaced by $|\frac{f(\sqrt{x})}{y} - \frac{f(y)}{\sqrt{x}}| > \frac{1}{2}$? | |
| Mar 2, 2011 at 8:45 | comment | added | Jose Arnaldo Bebita | Hi Joe, thanks for providing this construction. I am actually looking at an $f$ which is a bit 'chaotic', but I suspect it's injective over a particular subset of the natural numbers. | |
| Mar 2, 2011 at 8:10 | history | answered | Joe | CC BY-SA 2.5 |