Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Questions on Continuity

  1. May 13, 2013 #1
    ##prop:## let set ##E \subset \mathbb{R}## be unbounded, then ##\forall f## well-defined on ##E##, if ##f## is continuous, then ##f## is uniformly continuous.

    First am I reading this correctly, and second, I am having a hard time seeing this. Could someone please shed some light on this?

    Thanks.
     
  2. jcsd
  3. May 13, 2013 #2
    surely you meant bounded instead of unbounded, right?
     
  4. May 13, 2013 #3
    If it was bounded, then ##E## must be closed as well for ##f## to be uniform continuous. I am citing a case where ##E## is

    not bounded. Rudin gives the example of ##\mathbb{Z}## and states that ANY function defined on ##\mathbb{Z}## is

    indeed uniformly continuous.
     
  5. May 13, 2013 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Is your definition of unbounded weird? Because if E=R then it's saying that all continuous functions are uniformly continuous.

    The example of Z isn't because Z is Z is unbounded, the key property is that Z is discrete - any discrete set has that all functions on them are uniformly continuous.
     
  6. May 14, 2013 #5
    Yea I agree. That is why I am asking. Please see the attached theorem 4.20. The assumption on the boundedness is at the end of page 2
     

    Attached Files:

  7. May 14, 2013 #6

    lavinia

    User Avatar
    Science Advisor

    the theorem says there exists a continuous function on E that is notuniformly continuous
     
  8. May 14, 2013 #7
    Maybe I am reading too much into this.

    After equation (23), Rudin writes:

    "...Assertion (c) would be false if boundedness were omitted from the hypotheses."

    Can you explain this further? especially via an example without using the set of integers.
     
  9. May 17, 2013 #8
    He means there are some noncompact unbounded sets E for which all continuous functions on E are uniformly continuous. Of course, any unbounded set is noncompact, so he is saying that there are some unbounded sets E for which all continuous functions on E are uniformly conitnuous.

    By the way, note the word "some". That's why the proposition in your OP isn't stated correctly.
     
  10. May 17, 2013 #9

    Bacle2

    User Avatar
    Science Advisor

    It is definitely not always true, if I understood correctly ( or, If I have not jumped the gun, like I have sometimes done, embarrassingly).

    Take f: Q<ℝ → Q , with f(x)=1/(x-√2) .

    Q is unbounded in ℝ , but f is not uniformly-continuous (fails near √2 ; if you want it to fail
    at more points, you can repeat the idea.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook