Results 1 to 4 of 4

Math Help - Continuity Question

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    56

    Continuity Question

    Discuss the continuity of the function defined by:

    f(x)=\begin{cases}<br />
 & x\text{ if } x\,\in\,Q \\ <br />
 & -x\text{ if } x\,\notin \,Q <br />
\end{cases}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602
    Quote Originally Posted by vuze88 View Post
    Discuss the continuity of the function defined by:

    f(x)=\begin{cases}<br />
 & x\text{ if } x\,\in\,Q \\ <br />
 & -x\text{ if } x\,\notin \,Q <br />
\end{cases}
    Every rational number lies between two irrational numbers, and vice versa.

    So the function is discontinuous everywhere...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    56
    Can you explain why every rational number is between two irrational numbers? Also, according to the textbook, it is discontinuous everywhere but is continuous at 0.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,396
    Thanks
    1846
    Yes, the function is continuous at 0. Given any \epsilon> 0, [tex]|f(x)- f(0)| is |x- 0|= |x| if x is rational, and |-x- 0|= |x| if x is irrational. Take \delta to be any number less than \epsilon.

    If a is any rational number other than 0, f(x)= a so |f(x)- f(a)= |x-a| if x is rational and |-x-a|= |x+a| if x is irrational. No matter how small \delta is, there will exist both rational and irrational numbers between a-\delta and a+ \delta so with \epsilon less than half the difference between them, there will exist x such that |f(x)- f(a)|> [tex]\epsilon[/b]. The same is true if a is irrational.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuity question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 3rd 2010, 11:52 AM
  2. Continuity question
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: May 22nd 2010, 03:38 PM
  3. Continuity Question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: June 6th 2009, 10:09 PM
  4. Continuity question.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 27th 2008, 06:12 PM
  5. Continuity Question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 2nd 2007, 02:39 PM

Search Tags


/mathhelpforum @mathhelpforum