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December 4th, 2009, 10:02 PM   #1
 
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Continuity of multivariable functions

For what values of the number is the following function continuous on ?

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December 5th, 2009, 04:47 PM   #2
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Re: Continuity of multivariable functions

Your latex expressions aren't working!
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December 5th, 2009, 08:17 PM   #3
 
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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
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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.
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December 7th, 2009, 01:25 AM   #5
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The reason why latex was not interpreted was that bigli had selected the "Disable BBCode" option.
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December 10th, 2009, 09:08 PM   #6
 
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Re: Continuity of multivariable functions

For what values of the number is the following function continuous on ?

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