Results 1 to 2 of 2

Math Help - Uniform Continuity and Continuity

  1. #1
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Uniform Continuity and Continuity

    Let S be the set of points on an interval and let f be continuous on S. Then at every point x there is a neighborhood of x st

    |f(x)-f(y)|<ε if |x-y|<δ(x).

    The set of all δ(x) form an open cover of S. If S is compact, (closed and bounded, [a,b]), then there is a finite collection of the δ(x) which covers S. Then δ = min δi(x) and f is uniformly continuous. That establishes sufficiency. Necesssity established by uniform continuity implies continuity.

    If S is not bounded, we have to choose a minimum from an infinite collection of δ(x). If you show that δ(x) < δ for all x, you havenít determined that a min δ exists for all x and you havenít proved uniform continuity.*

    Conclusion:
    1) f(x) continuous on S compact ([a,b]) iff f(x) uniformly continuous.
    2) f(x) continuous on [a,∞) and lim f(x)=k as x→∞, does not imply uniform continuity.

    *Given all the δ(x) < δ, is there a minimum one(s) among them- a subtle point. For example, if you can show for a given ε, 0 < δ(x) < α, what is the minimum value of δ(x)?

    EDIT: There is an ambiguity in the definition of Uniform Continuity, depending on whether, given ε for the interval, you require:
    3) A SPECIFIC MIN δ(ε), or
    4) ANY δ LESS THAN δ(ε)

    The conclusions 1) and 2) above assume definition 3).
    Last edited by Hartlw; February 22nd 2014 at 08:14 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: Uniform Continuity and Continuity

    Conclusion 2) in OP is incorrect, and everything past 1), and the sentence previous to 1), should be discarded.

    Revised Conclusion:

    1í) f(x) continuous on S compact ([a,b]) iff f(x) uniformly continuous.
    2í) f(x) continuous on [a,∞) and lim f(x)=k as x→∞, implies uniform continuity.

    A proof of 2) is given by slipeternal in post #19 of
    Uniform continuity

    A clarified and expanded version is given below. If the expansion is part of the standard proof, then my apologies for posting it Peer Math Review, and this post will just be a clean-up of the previous one.

    1) Let b be smallest x st |f(x) - k│ < ε/4, x>b. Then
    |f(x) - f(y)│ < ε/2 for all │x - y│in [b,∞). Because lim f(x)=k as x →∞.

    2) There is a δ such that |f(x) - f(y)│ < ε/2 for │x - y│< δ, x,y in[a,b]. Heine-Borel.

    3) If x<b and y>b, |f(x) - f(y)│≤ |f(x) - f(b)│+ |f(y) - f(b)│< ε/2+ ε/2 = ε for │x - b │< δ and all y>b, or │x - y│< δ because │x - y│< δ → │x - b │< δ.

    4) Suppose bí>b. Then (abbreviated)
    a) |f(x) - f(y)│ < ε/2 for all │x - y│in [bí,∞)
    b) |f(x) - f(y)│< ε/2 for all │x - y│in [b,bí].
    c) |f(x) - f(y)│< ε/2 for │x - y│< δ all x,y in [a,b].
    So
    |f(x) - f(y)│< ε for │x - y│< δ for all x,y in [b,∞).

    The last step is very important because it shows δ depends on a single b.

    So continuity implies uniform continuity is consistent for the both conditions, on [a,b], and on [a,∞) with limf(x)=k.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniform continuity help
    Posted in the Calculus Forum
    Replies: 6
    Last Post: March 1st 2011, 12:54 PM
  2. Uniform Continuity
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 13th 2010, 12:50 PM
  3. Uniform Continuity
    Posted in the Calculus Forum
    Replies: 12
    Last Post: January 24th 2010, 07:25 AM
  4. Continuity, Uniform Continuity
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 1st 2009, 09:36 PM
  5. uniform continuity
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 18th 2008, 03:44 PM

Search Tags


/mathhelpforum @mathhelpforum