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 December 4th, 2009, 10:02 PM #1 Member   Joined: Dec 2006 Posts: 90 Thanks: 0 Continuity of multivariable functions For what values of the number $a$ is the following function continuous on $\Bbb{R}^3$? $\left \{\begin{array}{ll}\displaystyle{\frac{(x+y+z)^a}{ (x^2+y^2+z^2)}} \ (x,y,z)\not= 0\\0 \ (x,y,z)= 0 \ . \end{array}\right.$
 December 5th, 2009, 04:47 PM #2 Global Moderator   Joined: May 2007 Posts: 6,378 Thanks: 542 Re: Continuity of multivariable functions Your latex expressions aren't working!
 December 5th, 2009, 08:17 PM #3 Member   Joined: Dec 2006 Posts: 90 Thanks: 0 Re: Continuity of multivariable functions For what values of the number a is the following function continuous on R^3? f(x,y,z)=((x+y+z)^a)/(x^2+y^2+z^2) if (x,y,z)=/ 0 f(x,y,z)=0 if (x,y,z)= 0
December 6th, 2009, 02:27 PM   #4
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Re: Continuity of multivariable functions

Quote:
 Originally Posted by bigli For what values of the number a is the following function continuous on R^3? f(x,y,z)=((x+y+z)^a)/(x^2+y^2+z^2) if (x,y,z)=/ 0 f(x,y,z)=0 if (x,y,z)= 0
As you must have noticed there is only one point in question - x=y=z=0.

If a is an integer, then a>2 is obviously needed for continuity. At a=2, it looks like the expression is undefined.
Examples: x=y=0, let z->0 then f -> 1. z=0, x=y, let x -> 0 then f-> 2.

I suspect for a > 2, non-integer, you will have continuity, while for a < 2, f will blow up. You might try L'Hopital's rule to prove these.

 December 7th, 2009, 01:25 AM #5 Global Moderator   Joined: Dec 2006 Posts: 18,154 Thanks: 1422 The reason why latex was not interpreted was that bigli had selected the "Disable BBCode" option.
 December 10th, 2009, 09:08 PM #6 Member   Joined: Dec 2006 Posts: 90 Thanks: 0 Re: Continuity of multivariable functions For what values of the number $\alpha$ is the following function continuous on $R^3$ ? $f(x,y,z)=\left \{\begin{array}{cc}\frac{(x+y+z)^{\large{\alpha}}} {(x^2+y^2+z^2)} &\ \ \mbox{if}\ (x,y,z)\not= 0 ,\\0\ &\ \mbox{if}\ (x,y,z)= 0 \ . \end{array}\right.$

### continuity of multi

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