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Math Help - Continuous function of 2 variable

  1. #1
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    Thumbs up Continuous function of 2 variable

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

    for \forall y_{0}

    the limitation:

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

    exists.

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

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

    show that:

    g(x,y) is continuous on \{(x,y) | x\geq0, y\in\mathbb{R}\}
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Xingyuan View Post
    Assume f(x,y) is continuous on \{(x,y) | x>0, y\in\mathbb{R}\}

    for \forall y_{0}

    the limitation:

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

    exists.

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

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

    show that:

    g(x,y) is continuous on \{(x,y) | x\geq0, y\in\mathbb{R}\}
    Where are you stuck?
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