1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Construct a counterexample

  1. Jan 23, 2006 #1
    I was asked to show whether this is true: f(x) is defined for all x in [a,b] with f(b) > f(a) [values given]. the values of f at any x in (a,b) is rational. So, is f(x) continous?

    I think this is not continuous as this seems like the question is trying to use intermediate value property to imply continuity. But I can;t think of a more proper way to proof the answer. Or am I wrong on this? Please give me some idea!
  2. jcsd
  3. Jan 24, 2006 #2
    Construct a counterexample.
  4. Jan 24, 2006 #3


    User Avatar
    Science Advisor

    "this seems like the question is trying to use intermediate value property to imply continuity." I would have said it the other way around! If the function were continuous, then it would have the intermediate value property. Whatever f(a) and f(b) are, since f(b)> f(a) they are not the same. Does there exist an irrational number between them? What does that tell you?
  5. Jan 24, 2006 #4
    Right. It's not continuous. You can prove there is (are) at least an irrational number between f(a) and f(b).

    Let's use [tex]\sqrt 2 < 2[/tex] as the irrational number. Put A = f(a) and B = f(b). You can find a natural number n which satisfies [tex] B - A < \frac 1 n[/tex] Then, at least one of[tex]\sqrt 2 \frac m {2n} [/tex] (where m is an integer) must be between A and B. (notice step of [tex]\sqrt 2 \frac m {2n}[/tex] is smaller than B-A.)

    Then you can prove f(x) isn't continuous : If f(x) is continuous, it must pass such a [tex]\sqrt 2 \frac m {2n}[/tex] which is evidently an irrational number.

    BTW you can easily prove the opposite theory: If f(x) takes only irrational numbers between [a,b], then f(x) cannot be continuous.
    Last edited: Jan 24, 2006
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook