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Math Help - Continuity of multivariable function

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    Continuity of multivariable function

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


    f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a  lpha}}<br />
{(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 \alpha is the following function continuous on R^3 ?


    f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a  lpha}}<br />
{(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

     <br />
x = \rho \cos \theta \sin \phi<br />
     <br />
y = \rho \sin \theta \sin \phi<br />
     <br />
z = \rho \cos \phi<br />

    Then

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