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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Continuous functions with everywhere infinite variation with respect to sequences


Author: Z. Buczolich
Journal: Proc. Amer. Math. Soc. 103 (1988), 497-502
MSC: Primary 26A15; Secondary 26A45
MathSciNet review: 943073
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Abstract: We prove that if $ \{ {a_n}\} _{n = 1}^\infty $ is such that $ {a_n} \searrow 0$ and

$\displaystyle \mathop {\lim }\limits_{n \to \infty } ({a_n}_{ + 1}/{a_n}) = 1,$

then for the typical continuous function $ f$ we have

$\displaystyle {S_{{n_0}}}: = \sum\limits_{n = {n_0}}^\infty \vert f({x_n}_{ + 1}) - f({x_n})\vert = + \infty $

whenever $ x \in [0,1 - {a_{{n_0}}}]$ and $ {x_n} \in [x + {a_{n + 1}},x + {a_n}]$.

Based on our result in a previous paper, we know that the above theorem fails to hold if $ {a_{n + 1}}/{a_n} = \lambda < 1$.

We also prove that if $ \{ {a_n}\} _{n = 1}^\infty $ is such that $ {a_n} \searrow 0$, then for the typical continuous function $ f$ we have $ {S_{{n_0}}} = + \infty {\text{ if }}{x_n} = x + {a_n}$ and $ x \in [0,1 - {a_{{n_0}}}]$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0943073-4
PII: S 0002-9939(1988)0943073-4
Article copyright: © Copyright 1988 American Mathematical Society