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Real Analysis: Coninuity2

  1. Nov 12, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that f(x) = 3*x + 11 is uniformly continuous.

    2. Relevant equations

    x,y in S and |x-y| < a imply |f(x)-f(y)| < e

    3. The attempt at a solution

    following the book I get to

    f(x) - f(y) = 3(x-y)

    i just dont know how to chose e in this case..

    would it be |f(x) - f(y)| < e/3

    any pointers are greatly appreciated
  2. jcsd
  3. Nov 12, 2008 #2


    Staff: Mentor

    The idea is that you can make f(a) arbitrarily close to f(b) by specifying how close a must be to b, regardless of where a and b are.

    So if you wanted f(a) to be within 0.03 of f(b), how close must a be to b?
  4. Nov 12, 2008 #3
    You already did, you just forgot to say that the proof was done. By the time you get to | f(x) - f(y) | = | 3(x-y) | then look, you know you can make (x-y) smaller than some d, how small? Smaller than e/3. Then it follows that for e>0
    if |(x-y)| < d = (e/3) then |3(x-y)| < 3*(e/3) = e
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