Results 1 to 2 of 2

Thread: continuety of a complex function question..

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    continuety of a complex function question..

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by transgalactic View Post
    What do you mean by complex function? I assme you mean that a difficult function...I assume this because the sign function is only defined on the reals.

    1. f(g(x))=1+\text{sign}(x)^2

    Note that \text{sign}(x)=\left\{\begin{array}{rcl} -1 & \mbox{if} & x<0 \\ 0 & \mbox{if} & x=0\\ 1 & \mbox{if} & x>0 \end{array} \right.

    So 1+\text{sign}(x)^2=\left\{ \begin{array}{rcl} 2 & \mbox{if} & x\ne0\\ 1 & \mbox{if} & x=0 \end{array} \right.

    So f(g(x)) is continuous everywhere except zero.

    g(f(x))=\text{sign}\left(1+x^2\right). Now since 1+x^2>0\implies \text{sign}\left(x^2+1\right)=1~~\forall x \in\mathbb{R}


    2. f(g(x))=\sin(\ln(x))

    Now it is known that the composition of two continuous functions is continous. So at all points of definition this function is continuous so it is continuous for all values of x>0

    g(f(x))=\ln(\sin(x))

    Same concept here except we have that the function is defined on the set S=\left\{x:\sin(x)>0\right\}

    3. f(g(x))=\lfloor \text{sign}(x)\rfloor

    Notice that this changes nothing since \text{sign}(x)\in\mathbb{Z}

    g(f(x))=\text{sign}\left(\lfloor x\rfloor\right)

    Note that x<0\implies \lfloor x\rfloor<0


    0\leqslant x<1\implies \lfloor x\rfloor=0

    x\geqslant 1\implies \lfloor x\rfloor>0
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. uuniform continuety question
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Sep 3rd 2011, 01:11 PM
  2. integral/continuety proof question..
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 24th 2009, 10:16 AM
  3. continuety parameter question..
    Posted in the Calculus Forum
    Replies: 20
    Last Post: Jan 22nd 2009, 02:34 PM
  4. continuety of splitted function..
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 29th 2008, 12:59 PM
  5. continuety prove question..
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Dec 27th 2008, 06:25 AM

Search Tags


/mathhelpforum @mathhelpforum