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Homework Help: Analysis Question -

  1. Feb 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose f:[a,b]->R is continuous and that f([a,b]) is a subset of Q (rational numbers). Prove that f is constant on [a,b]


    2. Relevant equations
    N/A


    3. The attempt at a solution
    The solution states that:

    f([a,b]) must contain only one point, because if it has more than that, then it would have to be an irrational point. Which is a contradiction.


    My questions are, why does f([a,b]) contain only one rational point? Why can't it be a collection of rational points?

    Thank you for your help.

    M
     
  2. jcsd
  3. Feb 11, 2010 #2

    Hurkyl

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    Because then f would be discontinuous! (assuming the statement is true, which it is)
     
  4. Feb 11, 2010 #3
    Remember that between any 2 rational points (in R) there is an irrational point.
     
  5. Feb 11, 2010 #4
    You guys are life savers.

    Thank you!

    M
     
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