# Continuity of multivariable function

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

$\displaystyle f(x,y,z)=\left \{\begin{array}{cc}\displaystyle{\frac{(x+y+z)^{\a lpha}} {(x^2+y^2+z^2)}} \ & \mbox{if}\ (x,y,z)\not= 0\\0 \ & \mbox{if}\ (x,y,z)= 0 \ . \end{array}\right.$
• Dec 11th 2009, 12:25 PM
Jester
Quote:

Originally Posted by bigli
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 $\displaystyle \alpha > 2$.