Continuous functions with everywhere infinite variation with respect to sequences
HTML articles powered by AMS MathViewer
- by Z. Buczolich PDF
- Proc. Amer. Math. Soc. 103 (1988), 497-502 Request permission
Abstract:
We prove that if $\{ {a_n}\} _{n = 1}^\infty$ is such that ${a_n} \searrow 0$ and \[ \lim \limits _{n \to \infty } ({a_n}_{ + 1}/{a_n}) = 1,\] then for the typical continuous function $f$ we have \[ {S_{{n_0}}}: = \sum \limits _{n = {n_0}}^\infty | f({x_n}_{ + 1}) - f({x_n})| = + \infty \] 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 $x \in [0,1 - {a_{{n_0}}}]$.References
- Z. Buczolich, For every continuous $f$ there is an absolutely continuous $g$ such that $[f=g]$ is not bilaterally strongly porous, Proc. Amer. Math. Soc. 100 (1987), no. 3, 485–488. MR 891151, DOI 10.1090/S0002-9939-1987-0891151-X
- P. Humke and M. Laczkovich, Typical continuous functions are virtually nonmonotone, Proc. Amer. Math. Soc. 94 (1985), no. 2, 244–248. MR 784172, DOI 10.1090/S0002-9939-1985-0784172-7
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 497-502
- MSC: Primary 26A15; Secondary 26A45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0943073-4
- MathSciNet review: 943073