Results 1 to 2 of 2

Thread: Continuous function of 2 variable

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    66

    Thumbs up Continuous function of 2 variable

    Assume $\displaystyle f(x,y)$ is continuous on $\displaystyle \{(x,y) | x>0, y\in\mathbb{R}\}$

    for $\displaystyle \forall$$\displaystyle y_{0}$

    the limitation:

    $\displaystyle \lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0})$

    exists.

    now we define function $\displaystyle g(x,y)$ as:

    $\displaystyle g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases}$

    show that:

    $\displaystyle g(x,y)$ is continuous on $\displaystyle \{(x,y) | x\geq0, y\in\mathbb{R}\}$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Xingyuan View Post
    Assume $\displaystyle f(x,y)$ is continuous on $\displaystyle \{(x,y) | x>0, y\in\mathbb{R}\}$

    for $\displaystyle \forall$$\displaystyle y_{0}$

    the limitation:

    $\displaystyle \lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0})$

    exists.

    now we define function $\displaystyle g(x,y)$ as:

    $\displaystyle g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases}$

    show that:

    $\displaystyle g(x,y)$ is continuous on $\displaystyle \{(x,y) | x\geq0, y\in\mathbb{R}\}$
    Where are you stuck?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. continuous random variable
    Posted in the Statistics Forum
    Replies: 3
    Last Post: Nov 5th 2010, 12:57 PM
  2. Replies: 1
    Last Post: Sep 3rd 2010, 10:51 PM
  3. a continuous random variable with probability density function
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: Dec 5th 2009, 09:15 AM
  4. continuous random variable help?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Nov 23rd 2009, 01:53 AM
  5. Continuous Random Variable and Density Function
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: Nov 2nd 2007, 05:57 AM

Search Tags


/mathhelpforum @mathhelpforum