Timeline for Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| May 23 at 14:02 | comment | added | Nate River | @GiorgioMetafune Oh, I think I misread his second statement as saying something else. It is true. I guess he wanted to use it to conclude a contradiction by Sard’s theorem. | |
| May 23 at 13:51 | comment | added | Giorgio Metafune | Why is the second statemnent of @KlashF's not true? | |
| May 23 at 11:04 | vote | accept | Nate River | ||
| May 23 at 10:21 | answer | added | Guido De Philippis | timeline score: 6 | |
| May 20 at 22:58 | comment | added | KhashF | @NateRiver Thanks for clarification. | |
| May 20 at 22:10 | comment | added | Nate River | @KhashF The first statement is indeed true, and Sard’s theorem says that almost every $t$ is a regular point. However Sard’s only applies for smooth enough functions, $C^n$ in this case. The second statement is not true as stated, though something similar does hold. The keywords are $C^1$ Sard’s lemma. | |
| May 20 at 14:24 | comment | added | KhashF | Isn't it true that if $t$ is a regular value, then $Df$ is never zero on $E_t$; and $E_t$ has a positive $\mathcal{H}^{n-1}$ measure because it is a $C^1$ submanifold of dimension $n-1$? | |
| May 20 at 9:49 | comment | added | Nate River | @FedorPetrov Indeed, in a later edit I added that additional remark. | |
| May 20 at 9:45 | comment | added | Fedor Petrov | For more smooth functions Sard's theorem says that for almost all $t$ we have $Df\ne 0$ on $E_t$ | |
| May 20 at 8:55 | history | edited | Nate River | CC BY-SA 4.0 |
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| May 20 at 3:44 | history | asked | Nate River | CC BY-SA 4.0 |