Timeline for Why do we teach calculus students the derivative as a limit?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2020 at 9:41 | comment | added | Jochen Wengenroth | $\exp(-1/x^2)$ is at the heart of all distribution theory. | |
| Nov 9, 2010 at 12:10 | comment | added | Pablo Lessa | Take $f(x) = \exp(-1/x^2)$ and $g$ a continuous nowhere differentiable function. The function $h(x) = f(x)g(x)$ is continuous, goes to zero faster then any polynomial when $x \to 0$ but isn't differentiable at any point other than $0$. Hence $h$ isn't twice differentiable. | |
| Nov 2, 2010 at 22:59 | comment | added | Steven Gubkin | @ Benoi: Are there examples of such functions which are continuous in a neighborhood of 0? I ask because the only examples I can find are along the lines of the characteristic function of the rationals time exp(-1/x^2). | |
| Oct 19, 2010 at 14:11 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Sep 27, 2010 at 16:57 | comment | added | Pablo Lessa | You're right! Thank you. To be honest, I didn't consider this possibility at the time of writing. I'll leave the parenthesis as is, since it still has some content and you're comment is right below. | |
| Sep 27, 2010 at 15:53 | comment | added | Benoît Kloeckner | The example is well-chosen, but your parenthesis sounds misleading to me :there exist functions that go to zero faster than any polynomial at zero, while they are not even twice derivable. | |
| Sep 27, 2010 at 10:03 | history | answered | Pablo Lessa | CC BY-SA 2.5 |