
Re: Functions asymptotic to the xaxis
Posted:
Jul 28, 2011 9:28 AM


In article <20110728.040300@whim.org>, Rob Johnson <rob@trash.whim.org> wrote: >In article <20110727220251.J48523@agora.rdrop.com>, >William Elliot <marsh@rdrop.com> wrote: >>Assume f:R > R and for all x, lim(n>oo) f(nx) = 0. >> >>If f is continuous, does lim(x>oo) f(x) = 0? >> >>Is continuity needed? > >The problem as stated is trivial by letting x = 1. Therefore, I >assume that you intend n to be an integer. > >Continuity is necessary. Let f be defined as > > k > f(n/?^k) =  > k + n/?^k > > f(x) = 0 elsewhere > >where ? is the golden ratio. f is welldefined and for >all x, lim_{n>oo} f(nx) = 0, where n is restricted to >the integers. > >If n = [k ?^k], then f(n/?^k) > 1/2 and n/?^k > k1. > >Thus, f(x) > 1/2 for arbitrarily large x = n/?^k. > >I haven't yet found what kind of continuity is necessary >to insure that lim_{x>oo} f(x) = 0.
Sorry. This doesn't show that continuity is necessary, but that there are discontinuous functions that satisfy the hypotheses but not the conclusion.
Rob Johnson <rob@trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font

