Results 1 to 2 of 2

Math Help - Real Analysis - Combinations of Continuous Functions #2

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    135

    Real Analysis - Combinations of Continuous Functions #2

    Assume h: R --> R is continuous on R and let K = {x: h(x) = 0}. Show that K is a closed set.

    Aside: (In the past we have used K to denote a compact set. I'm not sure if it is denoting one here or not. Would they have to specify in the beginning of the problem that K is compact?)
    Last edited by ajj86; November 21st 2008 at 09:04 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by ajj86 View Post
    Assume h: R --> R is continuous on R and let K = {x: h(x) = 0}. Show that K is a closed set.
    It is a closed set because it's the reciprocal image of a closed set by a cont. application.

    Proof :
    Let's show that the reciprocal image by a continuous application of an open set is open.
    Let x \in f^{-1}(O)=\{x \in R ~:~ f(x) \in O\}, where O is an open set. Because it's open, it's a neighbourhood of any of its point. Hence it is a neighbourhood of f(x).
    Because the function is continuous, f^{-1}(O) will be a neighbourhood of x.
    Hence \forall x \in f^{-1}(O), f^{-1}(O) is a neighbourhood of x. So it's a neighbourhood of all its points. So it's an open set.


    Now use Hausdorff's formula : f^{-1}(^cO)=^c f^{-1}(O)
    ^cO is a closed set.
    f^{-1}(O) is open, and therefore ^c f^{-1}(O) is a closed set.

    We're done.

    ~~~~~~~~~~~~~~~~~~~~~~~~
    K=h^{-1}(\{0\})
    and in the usual topology over R, {0} is a closed set.

    Aside: (In the past we have used K to denote a compact set. I'm not sure if it is denoting one here or not. Would they have to specify in the beginning of the problem that K is compact?)
    No, but they could ask you to show it is a compact.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. real analysis lipschitz continuous
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 26th 2009, 09:01 AM
  2. Real Analysis Continuous Functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 7th 2008, 08:16 AM
  3. Combinations of Continuous Functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 5th 2008, 02:15 PM
  4. Replies: 1
    Last Post: May 6th 2008, 10:45 PM
  5. Real Analysis continuous functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 29th 2007, 01:46 PM

Search Tags


/mathhelpforum @mathhelpforum