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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

For every continuous $ f$ there is an absolutely continuous $ g$ such that $ [f=g]$ is not bilaterally strongly porous


Author: Z. Buczolich
Journal: Proc. Amer. Math. Soc. 100 (1987), 485-488
MSC: Primary 26A46; Secondary 26A15
DOI: https://doi.org/10.1090/S0002-9939-1987-0891151-X
MathSciNet review: 891151
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Abstract: For any Darboux function $ f:\left[ {0,1} \right] \to {\mathbf{R}}$ and any $ 0 < \delta < 1$ there is a point $ x \in \left[ {0,1 - \delta } \right]$ and a sequence $ {x_n}$ such that

(a) $ {x_n} \in \left[ {x + {\delta ^{n + 1}},x + {\delta ^n}} \right]\left( {n = 1,2, \ldots } \right)$ and

(b) $ \sum\nolimits_{n = 2}^\infty {\left\vert {f\left( {{x_n}} \right) - f\left( {{x_{n - 1}}} \right)} \right\vert} < + \infty $.

Consequently, for every $ f \in C\left[ {0,1} \right]$ there is an absolutely continuous function $ g$ such that $ \left\{ {x:f\left( x \right) = g\left( x \right)} \right\}$ is not bilaterally strongly porous.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0891151-X
Article copyright: © Copyright 1987 American Mathematical Society