Question

f(x,y) = { (x^3y-xy^3)/(x^2+ y^2) , (x,y)cantbe0
{ 0 , (x,y)=(0,0)

Are the functions fx(x,y) and fy(x,y) continuous on R2? Is f differntiable at (0,0)?

Answer 1

Continuous means that the limit and the function output are the same. I.E.\[\Large \lim_{x\rightarrow a}f(x) = f(a)\]
Differentiable means you can differentiate at that point. I.E. \[\Large \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}\]This limit exists.

Answer 2

I think the point of this exercise is to get you used to the concept of limits on multi-variable functions.

Answer 3

As well as partial derivatives on multi-variable functions.

Answer 4

Well, its just a portion of a longer question ..
I said that it was continuous .. I've already calculated the partial derivitices for each, so after that, does that mean i just sub 0 in, and see if i get an answer.. or?

Answer 5

Then I'm supposed to calculate the second order mixed partial derv. fxy(x,y)(0,0) and fyx(x,y)(0,0)

Answer 6

Put in (0,0), see if you get an answer and see if the limit as the variables goes to (0,0) is the same.

Answer 7

ok, is the only way of seeing if the limit exists by doing the f(x+h, x) / h, i thought that was just finding the derivitive but a different way

Answer 8

and if your putting 0 into the denominator, isn't it not going to exist?

Answer 9

Is what not going to exist?

Answer 10

the limit.
ok, i dont know if i understand what your talking about ..

Answer 11

ok, is the only way of seeing if the limit exists by doing the f(x+h, x) / h, i thought that was just finding the derivitive but a different way