Continuity of multivariable functions

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Continuity of multivariable functions

For what values of the number $a$ is the following function continuous on $\Bbb{R}^3$?

$\left \{\begin{array}{ll}\displaystyle{\frac{(x+y+z)^a}{(x^2+y^2+z^2)}} \ (x,y,z)\not= 0\\0 \ (x,y,z)= 0 \ . \end{array}\right.$
bigli
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Re: Continuity of multivariable functions

mathman
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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
bigli
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Re: Continuity of multivariable functions

bigli wrote: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.
mathman
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The reason why latex was not interpreted was that bigli had selected the "Disable BBCode" option.
skipjack
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Re: Continuity of multivariable functions

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

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