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Prove that f must be a constant function

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data

    If f: ℝ → ℝ is a continuous function with the property that its range is contained in the set of integers, prove that f must be a constant function.

    2. Relevant equations

    3. The attempt at a solution

    I know why this is true, I just don't know how to begin an actual proof. So far I've thought of proving by contradiction, with letting f be discontinuous and use f(x) = [x], whose range set is contained in Z.

    I seem to have trouble with the format of a formal proof.
  2. jcsd
  3. Oct 3, 2011 #2
    OK, suppose f(x) and f(y) are not equal for some x and y. Now, you know that their difference must be at least 1, right? So, let [itex]\epsilon = 1[/itex] and...

    Do you see how this might work?
  4. Oct 3, 2011 #3
    It seems an epsilon-delta proof might work well here.
  5. Oct 3, 2011 #4


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    Homework Helper

    Are you allowed to use intermediate value theorem?
  6. Oct 3, 2011 #5
    Oh, ok I think I got it. Thanks.
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