(adsbygoogle = window.adsbygoogle || []).push({}); 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 ]

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# Homework Help: Limit continuous function of rational numbers

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