Continuity of multivariable function

• Dec 11th 2009, 11:10 AM
bigli
Continuity of multivariable function
For what values of the number $\alpha$ is the following function continuous on $R^3$ ?

$f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a lpha}}
{(x^2+y^2+z^2)}} \ & \mbox{if}\ (x,y,z)\not= 0\\0 \ & \mbox{if}\ (x,y,z)= 0 \ . \end{array}\right.$
• Dec 11th 2009, 01:25 PM
Jester
Quote:

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 $\alpha > 2$.