Continuity of multivariable function

**bigli** · December 11th 2009, 10:10 AM

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

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

**Danny** · December 11th 2009, 12:25 PM

Originally Posted by bigli

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

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

If you switch to spherical polar coords

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

Then

$ \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 $\alpha > 2$ .

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