Results 1 to 3 of 3

Thread: Compact continuity question.

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    48

    Compact continuity question.

    Prove that if $\displaystyle F\subset\mathbb{R}$, is not compact. Then there exists a continuous function $\displaystyle f:F\rightarrow\mathbb{R}$ which isn't bounded on $\displaystyle F$

    Any help would be appreciated.
    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
    10
    Quote Originally Posted by skamoni View Post
    Prove that if $\displaystyle F\subset\mathbb{R}$, is not compact. Then there exists a continuous function $\displaystyle f:F\rightarrow\mathbb{R}$ which isn't bounded on $\displaystyle F$
    If F is not compact then it is either unbounded or not closed (or both). If it is unbounded then (obviously) the function f(x) = x is unbounded on F. If F is not closed, let z be a point in the closure of F but not in F, and let f(x) = 1/(xz).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by skamoni View Post
    Prove that if $\displaystyle F\subset\mathbb{R}$, is not compact. Then there exists a continuous function $\displaystyle f:F\rightarrow\mathbb{R}$ which isn't bounded on $\displaystyle F$

    Any help would be appreciated.
    Two possibilities: (i) either F is not closed, (ii) F is not bounded.

    If F is not closed then there exists $\displaystyle p\in \partial F$ so that $\displaystyle p\not \in F$. Now define $\displaystyle f(x) = \tfrac{1}{|p-x|}$.
    This function is continous and unbounded since $\displaystyle x$ gets arbitrary close to $\displaystyle p$.

    If F is not bounded then for some $\displaystyle p\in F$ define $\displaystyle f(x) = |p-x|$.
    Certainly this is not bounded since $\displaystyle x$ gets arbitrary away from $\displaystyle p$.

    EDIT: Opalg beat me too it .

    By the way I think it is possible to find a $\displaystyle \mathcal{C}^{\infty}$ function , do you agree ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Nov 19th 2011, 06:32 AM
  2. Finite union of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Apr 8th 2011, 07:43 PM
  3. the intersection of a collection of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Feb 28th 2010, 01:58 PM
  4. Metric spaces - continuity / compact
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jun 29th 2009, 06:04 PM
  5. Replies: 2
    Last Post: Apr 6th 2007, 05:48 PM

Search Tags


/mathhelpforum @mathhelpforum