Results 1 to 4 of 4

Math Help - The collection of continuous point of a real-valued function

  1. #1
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374

    The collection of continuous point of a real-valued function

    Let f be a real-valued function defined for all real numbers, C be the set of points at which f is continuous, Show that C is a G_\delta set.
    Any help is appreciate !
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Shanks View Post
    Let f be a real-valued function defined for all real numbers, C be the set of points at which f is continuous, Show that C is a G_\delta set.
    Write B(x,r) for the open interval (xr,x+r). For n=1,2,3,..., define G_n = \{x\in\mathbb{R}:\exists\delta>0\text{ such that }f(B(x,\delta)\subseteq B(f(x),1/n)\}. Then G_n is open, so G = \textstyle\bigcap_{n=1}^\infty G_n is a G_\delta-set. Check that G is the set of points at which f is continuous.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374

    counterexample

    I did the same things as you, but I can't proof that each G_n is open.
    Here I give a counterexample that G_1 is not open :
    let f be defined as :
    f(0)=0;
    f(x)=\frac{3}{4} if x is in the set of rational numbers except 0;
    f(x)=-\frac{3}{4} if x is in the irrational numbers set.
    Then G_1=\{0\} is not open.
    Any way , thank you all the same !
    I've also got a new idea!
    I made a little change to G_n: By let
    G_n = \{x\in\mathbb{R}:\exists\delta>0\text{ such that }|f(x_1)-f(x_2)|\leq 1/n \text{ for any } x _1, x_2\in B(x,\delta)\}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Shanks View Post
    I made a little change to G_n: By let
    G_n = \{x\in\mathbb{R}:\exists\delta>0\text{ such that }|f(x_1)-f(x_2)|\leq 1/n \text{ for any } x _1, x_2\in B(x,\delta)\}
    You're quite right. I realised overnight that my sets G_n need not be open. Your modification above seems to repair that error.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Find a real-valued function
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 6th 2010, 08:53 PM
  2. Real Valued Function
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 22nd 2008, 03:01 AM
  3. Real-valued function
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 14th 2008, 08:10 AM
  4. Real valued function on [0,1]
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 15th 2008, 09:34 AM
  5. real valued function
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 26th 2008, 04:05 AM

Search Tags


/mathhelpforum @mathhelpforum