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Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

PS: Wouldn't it be funny if there was also a function that was Holder continuous of every order less than 1, but nowhere differentiable…

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

PS: Wouldn't it be funny if there was also a function that was Holder continuous of every order less than 1, but nowhere differentiable…

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Nate River
Nate River
  • 6.2k
  • 2
  • 23
  • 99

Can a differentiable function be nowhere locally $\alpha$-alpha HölderHölder for all $\alpha > 0$?

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-HolderHölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

Can a differentiable function be nowhere locally $\alpha$-alpha Hölder for all $\alpha > 0$?

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Holder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

Can a differentiable function be nowhere locally $\alpha$-Hölder for all $\alpha > 0$?

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

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Source Link
Nate River
Nate River
  • 6.2k
  • 2
  • 23
  • 99

A really non smooth Can a differentiable function be nowhere locally $\alpha$-alpha Hölder for all $\alpha > 0$?

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Holder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

A really non smooth differentiable function

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Holder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but I actually think this might be true…

Can a differentiable function be nowhere locally $\alpha$-alpha Hölder for all $\alpha > 0$?

Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Holder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but I actually think this might be true…

The answer by user479223 here shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.

Source Link
Nate River
Nate River
  • 6.2k
  • 2
  • 23
  • 99