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Continuous functions with everywhere infinite variation with respect to sequences

1988
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Proceedings of the American Mathematical Society
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We prove that if {onl^Lj is such that an \ 0 and lim (a"+i/on) = 1, n -oo then for the typical continuous function / we have oo Sn0 := Yl \f(xn + l) -f{xn)\ = +00 n = nn whenever x S [0,1 -ano] and x" € [x + a"+i, x + an}. Based on our result in a previous paper, we know that the above theorem fails to hold if an+i/an = A < 1. We also prove that if {an}^°=1 is such that an \ 0, then for the typical continuous function / we have Sno = +oo if xn = x + an and x € [0,1 -ano].

doi:10.1090/s0002-9939-1988-0943073-4
fatcat:pgttukmvlbgkxoz4p3foqfmfle