Results 1 to 4 of 4

Math Help - Continuity

  1. #1
    Banned
    Joined
    Sep 2009
    Posts
    502

    Continuity

    For function f to be continuous at a, three conditions must be satisfied:

    (1)  f is defined at a;
    (2) \lim_{x \to a} f(x) exists;
    (3) \lim_{x \to a} f(x)=f(a) exists;

    Now say a function g is defined as

    g(x) = \left \{\begin{array}{cc}0,&\mbox{if} x\not \in \mathbb{Z}\\1,&\mbox{if} x\in \mathbb{Z} \end{array}\right.

    Can we say g is continuous a where a=2 and a \in \mathbb{Z}?

    Does g satisfy the following?

    (1)  f is defined at a where f(2)=1;
    (2) \lim_{x \to a} f(x) exists;
    (3) \lim_{x \to a} f(x)=f(a)=0 exists

    ?
    Last edited by novice; August 2nd 2010 at 07:47 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member eumyang's Avatar
    Joined
    Jan 2010
    Posts
    278
    Thanks
    1
    Quote Originally Posted by novice View Post
    Now say a function g is defined as

    g(x) = \left \{\begin{array}{cc}0,&\mbox{if} x\not \in \mathbb{Z}\\1,&\mbox{if} x\in \mathbb{Z} \end{array}\right.

    Can we say g is continuous at a where a=2 and a \in \mathbb{Z}?

    Does g satisfies the following?
    (1)  g(a) is defined.
    Yes, g(2)=1.

    (2) \lim_{x \to a} g(x) exists;
    Yes, \lim_{x \to 2} g(x) = 0.

    (3) \lim_{x \to a} g(x)=g(a)
    No. We have in (1) that g(2)=1 and in (2) \lim_{x \to 2} g(x) = 0. They're not equal, so condition (3) isn't satisfied. So g is not continuous at x = 2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Sep 2009
    Posts
    502
    Now, change point a to a \not \in \mathbb{Z}.

    Then g(x) is contiunous at x= a,  \forall a \not \in \mathbb{Z}.

    Yah?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,792
    Thanks
    1687
    Awards
    1
    Yes indeed.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuity of a Sum on R
    Posted in the Calculus Forum
    Replies: 5
    Last Post: May 4th 2010, 08:40 AM
  2. Continuity
    Posted in the Calculus Forum
    Replies: 9
    Last Post: May 2nd 2010, 02:57 AM
  3. continuity
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 1st 2010, 09:09 PM
  4. Continuity, Uniform Continuity
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 1st 2009, 08:36 PM
  5. continuity
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 7th 2008, 11:22 PM

Search Tags


/mathhelpforum @mathhelpforum