# Multivariable Continuity

## 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.

## 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.

Related Calculus and Beyond Homework Help News on Phys.org
jbunniii
Homework Helper
Gold Member
The only term that can cause a problem is the last one:

$$\frac{4x^6y^2}{(x^4+y^2)^2}$$

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

$$\lim_{x,y \rightarrow 0}\frac{4x^6y^2}{(x^4+y^2)^2} = 0$$

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