A rough differentiable function

Kirchheim, Bernd; Müller, Paul

doi:10.1090/S0002-9939-08-09322-2

A rough differentiable function
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by Bernd Kirchheim and Paul F.X. Müller PDF

Proc. Amer. Math. Soc. 136 (2008), 3875-3881

Abstract:

A real-valued continuously differentiable function $f$ on the unit interval is constructed such that \[ \sum _{k=1}^\infty \beta _f (x, 2^{-k} ) = \infty \] holds for every $x \in [0,1].$ Here $\beta _f (x, 2^{-k} )$ measures the distance of $f$ to the best approximating linear function at scale $2^{-k}$ around $x$.

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