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A Holder Hölder continuous function which does not belong to any Sobolev space

I'm seeking a function which is Holder Hölder continuous but does not belong to any Sobolev space.

Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \in (0,1)$ and $\Omega$ a bounded set such that $u \notin W_{loc}^{1,p}(\Omega)$ for any $1 \leq p \leq \infty$. Take $\Omega$ to be bounded, open.

My first guess is to do a construction with a Weirstrass Weierstrass function. I know this is differentiable 'nowhere' but that doesn't convince me it isn't weakly differentiable in some bizarre way. Hopefully someone knows of an explicit example.
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I'm seeking a function which is Holder continuous but does not belong to any Sobolev space.

Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \in (0,1)$ and $\Omega$ a bounded set such that $u \notin W_{loc}^{1,p}(\Omega)$ for any $1 \leq p \leq \infty$. Take $\Omega$ to be bounded, open.

My first guess is to do a construction with a Weirstrass function. I know this is differentiable 'nowhere' but that doesn't convince me it isn't weakly differentiable in some bizarre way. Hopefully someone knows of an explicit example.
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A Holder continuous function which does not belong to any Sobolev space

I'm seeking a function which is Holder continuous but does not belong to any Sobolev space.

My first guess is to do a construction with a Weirstrass function. I know this is differentiable 'nowhere' but that doesn't convince me it isn't weakly differentiable in some bizarre way. Hopefully someone knows of an explicit example.