Multivariable Continuity

  • Thread starter Maraduke
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  • #1
Maraduke
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Homework Statement


Define f(0,0)=0 and f(x,y) = x2 +y2-2x2y-4x6y2/(x4+y2)2.

Show for all (x,y) that 4x4y2<=(x4+y2)2 and conclude that f is continuous.


Homework Equations





The Attempt at a Solution


Showing the inequality is trivial, but I do not see how I can conclude the function is continuous. I've done some messing around with the form of f, but am not getting anywhere.
 

Answers and Replies

  • #2
jbunniii
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The only term that can cause a problem is the last one:

[tex]\frac{4x^6y^2}{(x^4+y^2)^2}[/tex]

Since you have set the function equal to 0 at the origin, the following must be true in order to have continuity:

[tex]\lim_{x,y \rightarrow 0}\frac{4x^6y^2}{(x^4+y^2)^2} = 0[/tex]

Which part of this fraction has a [itex]4x^4y^2[/itex] in it? That's the logical place to try using the hint.
 

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