Timeline for Non-continuous higher differentiability, II
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 13, 2014 at 16:17 | comment | added | Mike Shulman | Yeah... but that's not enough, because the counterexamples $x^3 \sin(1/x)$ are smooth away from $0$. Maybe if one asserts that the value of that limit is a continuous function of $x$? | |
| May 12, 2014 at 23:57 | comment | added | Tom Goodwillie | No idea. Except that it seems clear that the statement in question must have a hypothesis about this limit existing for all nearby $x$, not just for one $x$. | |
| May 12, 2014 at 22:23 | comment | added | Mike Shulman | Indeed. Thanks! I am puzzled, however, because I just happened to be looking through Strichartz' The Way of Analysis, and on p181 after proving that $\lim_{h\to 0} \frac{1}{h^2} (f(x+2h)-2f(x+h)+f(x)) = f''(x)$ for a $C^2$ function, he says "There is also a converse to this theorem, but it is much more difficult to prove." Do you have any idea what he might have had in mind? | |
| May 12, 2014 at 12:40 | comment | added | Tom Goodwillie | All the examples given in the answers to your previous question ... | |
| May 11, 2014 at 16:32 | comment | added | Mike Shulman | Do you know an example that shows you can't phrase this definition using the quadratic form instead of the bilinear map? I.e. a differentiable function $f$ such that $f(x+2v)-2f(x+v)+f(x) = Q(v) + E(v) |v|^2$ for some quadratic form $Q$ and error $E(v)\to 0$, but $f$ is not twice differentiable? | |
| May 11, 2014 at 4:00 | vote | accept | Mike Shulman | ||
| May 11, 2014 at 2:31 | comment | added | Deane Yang | Oy. Stuff like this gives me a headache. It reminds me why analysts and geometers avoid studying differentiable functions per se. | |
| May 11, 2014 at 2:02 | comment | added | Tom Goodwillie | Do you believe that every $\mathbb Q$ vector space has a basis? | |
| May 11, 2014 at 1:59 | comment | added | Deane Yang | I understood it as $\mathbb{Q}$-linear, but does a nontrivial example exist? I'll settle for a proof that such an example exists. | |
| May 10, 2014 at 20:10 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
added 12 characters in body
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| May 10, 2014 at 20:09 | comment | added | Tom Goodwillie | I meant $\mathbb Q$-linear, sorry. I'll edit. But don't ask me for an example. | |
| May 10, 2014 at 19:41 | comment | added | Deane Yang | Really, really dumb question: what's a nontrivial linear map from $\mathbb{R}$ to $\mathbb{Q}$? | |
| May 10, 2014 at 12:59 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |