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Homework Help: Uniform Continuity proof, does it look reasonable?

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Note: I will use 'e' to denote epsilon and 'd' to denote delta.

    Using only the e-d definition of continuity, prove that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity).

    2. Relevant equations



    3. The attempt at a solution

    Proof:

    Must show that for each e>0 there is d>0 s.t.

    |x/(x+1) - a/(a+1)| < e whenever x,a are elements of [0, infinity) |x-a| < d.

    |x/(x+1) - a/(a+1)| = |(-x+a)/[(x+1)(a+1)]| [tex]\leq[/tex] |-x+a| = |x-a|.

    Thus, given e>0, if we choose d=e then,

    |x/(x+1) - a/(a+1)| < e whenever |x-a| < d.

    This implies that f(x) = x/(x+1) is uniformly continuous on [0,infinity). QED
     
  2. jcsd
  3. Mar 3, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure. That works. You could clean up few details, like x/(x+1) - a/(a+1)=(x-a)/((x+1)(a+1)), not (-a+x)/((x+1)(a+1)) and you could also explicitly justify why |(x-a)/((x+1)(a+1))|<=|x-a| but the proof works fine.
     
  4. Mar 3, 2010 #3
    Cool, thanks for the input.
     
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