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Homework Help: Uniform continuity

  1. Jan 30, 2010 #1

    1. The problem statement, all variables and given/known data
    Given that f is continuous in [tex][1,\infty)[/tex] and [tex] lim_{x->\infty}f(x) [/tex] exists and is finite, prove that f is uniformly continuous in [1,[tex]\infty)[/tex]

    3. The attempt at a solution
    We will mark [tex] lim_{x->\infty}f(x) = L [/tex]. So we know that there exists [tex]x_{0}[/tex] such that for every [tex] x>x_{0} |L-f(x)|<\epsilon [/tex] so f is uniformly continuos in [tex] (x_{0},\infty) [/tex]
    We shall look at the segment [tex] [1,x_{0}+1] [/tex]. We know that a continuous function ina closed segment is uniformly continuous.

    From here I am puzled. I have two routes.
    The easy route sais that since we showed f is uniformly continuous in those two segments and they overlap then f is uniformly continuous on the whole segment. I havea feeling this is not enough.
    Route 2:
    We know that a continous function ina closed segment has a minimum and maximal value there. Then we can look at the closed segment [tex] [1,x_{0}+1] [/tex] and mark [tex] m=MIN(f(x)), M= MAX(f(x)) [/tex]. But since f converges to L from [tex] x=,x_{0} [/tex] we have taken into account all of the possible values of f.
    we will mark [tex] d= | |M| - |m| | [/tex] the biggest difference in value f gets.
    and from here I am stuck. I am lost as to how to tie up the proof. Gudiance would be greatly apreciated.
  2. jcsd
  3. Jan 30, 2010 #2
    I think that I would try to use your "route 1", if you know that the mapping is uniformly continuous in both domains, and they overlap on [x0,x0+1]. Let's call them domains 1 and 2.

    Uniform continuity in a domain demands that for any constant epsilon>0, there should be a constant delta>0 such that |f(y)-f(x)| < epsilon for all y with |y-x| < delta.

    For a given epsilon, you can find such a delta for domain 1, and one for domain 2, just choose the smallest one of those, and I think is it OK, right? I guess delta also has to be less than the overlap of the two domains. You can always choose such a delta.

    It's been a while since I worked on such problems, so please be very critical towards my statements!

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