1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    gb7nash

    User Avatar
    Homework Helper

    Are you allowed to use intermediate value theorem?
     
  6. Oct 3, 2011 #5
    Oh, ok I think I got it. Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook