Continuous functions with everywhere infinite variation with respect to sequences

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 $x \in [0,1 - {a_{{n_0}}}]$.