# Thread: Two variable function hard problem

1. ## Two variable function hard problem

Please if you can solve this problem :S
http://www13.0zz0.com/2009/11/19/17/395971290.jpg << Continuous not Constant

2. It's clearly not constant (take (1,0) and (0,1)). Don't you mean continous?

3. Exactly what am thinking of =)
but i thought the definition of a multivariable function is different ..

Sorry..
continuous not constant xD ..
and its my friend's question ;>

4. Just notice by the triangle inequality $\displaystyle \vert x\frac{x^2-y^2}{x^2+y^2} \vert \leq \vert x\frac{x^2+y^2}{x^2+y^2} \vert = \vert x \vert$ so take $\displaystyle \vert x \vert \leq \Vert (x,y) \Vert \leq \delta = \epsilon$ and plug it in the definition of continuty. This gives you that the function is continous at 0, for every other point it's clearly continous since it's the quotient of cont. functions and the denominator does not vanish in any other point.

5. Thanks ..
It can be solved by algebra?
I mean without using the definition of limit .. ?