Results 1 to 3 of 3

Math Help - Continuous Functions

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    80

    Continuous Functions

    Let f:\mathbb{R} \rightarrow \mathbb{R} be continuous at c and let f(c)>0. How would you show that there is a neighborhood V_{\delta}(c) of c such that if x \in V_{\delta}(c), then f(x) > 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Feb 2010
    Posts
    147
    Forgive my lack of syntax here

    if f is continuous at c, then for e> 0 there exists a d>0 st

    0< l x - c l < d --> l f(x) - f(c) l < e

    but l f(x) - f(c) l < e is the same thing as:

    f(c) - e < f(x) < f(c) + e

    so as long as 0 < e < f(c) we have f(x) > 0 for all 0 < l x - c l < d

    and the conclusion follows
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by CrazyCat87 View Post
    Let f:\mathbb{R} \rightarrow \mathbb{R} be continuous at c and let f(c)>0. How would you show that there is a neighborhood V_{\delta}(c) of c such that if x \in V_{\delta}(c), then f(x) > 0
    The exact same concept as southprkfan1 but using open ball notation. Since \{x_:x>0\}=(0,\infty) is an open set we know there exists some B_{\varepsilon}(f(x))\susbseteq(0,\infty) but by continuity there exists some B_{\delta}(x) such that f\left(B_{\delta}(x)\right)\subseteq B_{\varepsilon}(x)\subseteq(0,\infty). The conclusion follows.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Set of all C-U continuous functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 27th 2010, 10:33 AM
  2. Continuous Functions!
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 20th 2010, 01:02 PM
  3. Continuous functions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 3rd 2009, 09:02 AM
  4. Continuous functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 1st 2009, 11:48 AM
  5. Continuous functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 2nd 2008, 01:18 AM

Search Tags


/mathhelpforum @mathhelpforum