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Limit continuous function of rational numbers

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Fix a real number a>1. If r=p/q is a rational number, we define a^r to be a^(p/q). Assume the fact that f(r)=a^r is a continuous increasing function on the domain Q of rational numbers r.

    Let s be a real number. Prove that lim r--->s f(r) exists

    2. Relevant equations
    For all ε>o there exists δ>0 s.t. |x-x0|<δ implies |f(x)-f(x0)|<ε
    Where x0 is an accumulation point for the domain

    3. The attempt at a solution

    Fix ε>0 and set δ=ε [not sure about this]
    With s an accumulation point for Q
    |r-s|<δ implies |f(r)-f(s)|<ε

    [trouble here]
    |f(r)-f(s)|= |a^r - a^s|
    [i'm not sure how I should alter this equation to get to |r-s|<δ=ε. Perhaps I am taking the wrong approach? I just need to show that a limit exists as r--->s ]
  2. jcsd
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