# Limit continuous function of rational numbers

1. Oct 14, 2012

### Krovski

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