Timeline for Can a nowhere locally Hölder function be differentiable almost everywhere?
Current License: CC BY-SA 4.0
12 events
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| Dec 9, 2023 at 14:45 | history | edited | user479223 | CC BY-SA 4.0 |
added 202 characters in body
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| Dec 8, 2023 at 23:37 | comment | added | user479223 | @NateRiver By the way, this gives an example of a continuous, increasing, bounded function with arbitrarily bad local modulus of continuity. Just replace $\log$ with $\log\log$ or even more slowly diverging functions. Differentiable a.e. but arbitrarily bad. | |
| Dec 8, 2023 at 20:06 | history | edited | LSpice | CC BY-SA 4.0 |
Holder -> Hölder
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| Dec 8, 2023 at 18:22 | comment | added | user479223 | The theorem about reparameterization uses the variation. The variation of an increasing function is always the function itself. So it is the inverse of the function. So stupid. I love math. | |
| Dec 8, 2023 at 18:20 | comment | added | user479223 | @NateRiver hah. What a stupid answer. Beautiful | |
| Dec 8, 2023 at 18:19 | comment | added | Nate River | I was just thinking about that. Actually, how bad is the reparametrization allowed to be? Because since $f$ is increasing continuous, we can reparametrize by its inverse and it just becomes the identity, which is Lipschitz. | |
| Dec 8, 2023 at 18:18 | comment | added | user479223 | @NateRiver Actually this is a really nice counterexample for something else too. Everyone knows that $\alpha$ Holder implies finite $1/\alpha$ variation. However finite $p$ variation doesn't imply $1/p$ Holder - only after a reparameterization. This is bounded variation but nowhere locally $\alpha$ Holder. I wonder what the reparameterization is... | |
| Dec 8, 2023 at 18:06 | comment | added | user479223 | @NateRiver Continuous, increasing, bounded but nowhere locally $\alpha$ Holder for any $\alpha$. What an awful function. | |
| Dec 8, 2023 at 18:03 | vote | accept | Nate River | ||
| Dec 8, 2023 at 18:03 | comment | added | Nate River | Very nice! The idea to use logs is great. | |
| Dec 8, 2023 at 18:00 | history | edited | user479223 | CC BY-SA 4.0 |
added 16 characters in body
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| Dec 8, 2023 at 17:53 | history | answered | user479223 | CC BY-SA 4.0 |