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Thread: Continuity of multivariable function

  1. #1
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    Continuity of multivariable function

    For what values of the number $\displaystyle \alpha$ is the following function continuous on $\displaystyle R^3$ ?


    $\displaystyle f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a lpha}}
    {(x^2+y^2+z^2)}} \ & \mbox{if}\ (x,y,z)\not= 0\\0 \ & \mbox{if}\ (x,y,z)= 0 \ . \end{array}\right. $
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  2. #2
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    Quote Originally Posted by bigli View Post
    For what values of the number $\displaystyle \alpha$ is the following function continuous on $\displaystyle R^3$ ?


    $\displaystyle f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a lpha}}
    {(x^2+y^2+z^2)}} \ & \mbox{if}\ (x,y,z)\not= 0\\0 \ & \mbox{if}\ (x,y,z)= 0 \ . \end{array}\right. $
    If you switch to spherical polar coords

    $\displaystyle
    x = \rho \cos \theta \sin \phi
    $
    $\displaystyle
    y = \rho \sin \theta \sin \phi
    $
    $\displaystyle
    z = \rho \cos \phi
    $

    Then

    $\displaystyle
    \lim_{\rho \to 0} \frac{\rho^{\alpha}\left( \cos \theta \sin \phi + \sin \theta \sin \phi + \cos \phi\right)^{\alpha}}{\rho^2}
    $ and for this limit to exist we need $\displaystyle \alpha > 2$.
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