Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By HallsofIvy

Math Help - What am I missing? (continuity)

  1. #1
    Newbie
    Joined
    Jul 2013
    From
    USA
    Posts
    9

    What am I missing? (continuity)

    Munkres "Analysis on Manifolds "P. 30 #5 asks: let f: X- > Y. Show that f is continuous if and only if for each x [in] X there is a neighborhood U of x such that flU is continuous. I can understand I think how to prove the condition is necessary. But since x is in U doesn't the condition state "f is continuous for all x in X?" (so why need to prove the condition is sufficient?)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What am I missing? (continuity)

    Hey krcmd.

    What do you mean by f|U? Is this f \ U?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,846
    Thanks
    321
    Awards
    1

    Re: What am I missing? (continuity)

    I don't know the notation well but it seems that f|U should be f: U --> Y: f(U)? Is this the meaning for f|_U?

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,393
    Thanks
    1327

    Re: What am I missing? (continuity)

    Yes, that is how I would interpret f|U: "f restricted to U".

    krcmd, the statement "f is continuous (on X) if and only if for each x [in] X there is a neighborhood U of x such that flU is continuous"
    has, of course, two parts:
    1) If, for each x in X, there is a neighborhood U of x such that f|U is continuous then f is continuous in X".
    That is the "meat" of the theorem. And
    2) If f is continuous in X, then there is a neighborhood U of x such that f|U is continuous".
    Yes, this follows immediately from the fact that f is continuous on X. There is a technical detail to deal with: strictly speaking, "f|U" is a different function from f.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2013
    From
    USA
    Posts
    9

    Re: What am I missing? (continuity)

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jul 2013
    From
    USA
    Posts
    9

    Re: What am I missing? (continuity)

    How is this reasoning? Form the union of all the open sets U within which f is continuous; the union U' is also open. X-U' consists of the boundary and the exterior of U'. Neither the boundary nor the exterior have any points with a neighborhood U where f is continuous. So X-U' is null, and X=U'

    I still have the nagging feeling that I'm missing something very important here.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: February 16th 2013, 06:38 AM
  2. I'm missing something
    Posted in the Statistics Forum
    Replies: 7
    Last Post: January 28th 2010, 05:45 AM
  3. Another p-value-am I missing anything?
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 14th 2009, 09:41 PM
  4. What am I missing here?
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 18th 2007, 03:57 PM
  5. Missing something
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: March 3rd 2007, 07:33 PM

Search Tags


/mathhelpforum @mathhelpforum