# Stuck in assignment due tomorrow(in fact this morning)

1. Sep 21, 2006

### precondition

I don't know if there's anyone who bother to answer my questions but still let me try..
If function f:R^2-->R is f(x,y)=0 if (x,y)=(0,0) and xy^2/(x^2+y^2) otherwise, then prove that f is not continuous at (0,0)
Supposed to be easy but my brain is dead right now for this..
it's multivariable analysis question by the way(proof has to be rigorous..)

2. Sep 21, 2006

### StatusX

It's sufficient to find a sequence (x_n,y_n) tending to (0,0) but with f(x_n,y_n) not tending to f(0,0)=0. Consider what the function looks like when one of x or y is very small compared to the other.

3. Sep 21, 2006

### precondition

Consider what the function looks like when one of x or y is very small compared to the other.

hmm....

4. Sep 21, 2006

### precondition

Got it!!!
let (x,y)=(1/n_2,1/n) then (x,y)-->(0,0) but f(x,y)-->infinity!! Thanks

umm... I have 6 more to go...
next one, if I'm allowed to ask more than 1 question, is
using same function as above,
Show directional derivative Dv(f)(x,y) exists for all (x,y) in R_2 and for all vector v in R_2. Then Calculate Dv(f)(x,y).
Now, first part I'm not quite sure, second part, can I use the fact that
Dv(f)(x,y)=sum over partial derivatives(with coefficients)

5. Sep 21, 2006

### StatusX

Wait, sorry, I think the original function is continuous. Are you sure you copied the question correctly?

6. Sep 21, 2006

### precondition

no it's not continuous and I proved it using the method you outlined..?!?!

7. Sep 21, 2006

### StatusX

If that's supposed to be $(x_n,y_n)=(1/n^2,1/n)$, then $f(x_n,y_n)=(1/n^4)/(1/n^2+1/n^4)=1/(n^2+1) \rightarrow 0$. Maybe x,y can be complex, or you need to prove it is continuous?

Last edited: Sep 21, 2006
8. Sep 21, 2006

### precondition

umm... its (1/n_4)/(1/n_4!!+1/n_4)=n_2 giving infinity...
Anyway could you please....... answer my next question plz... I'm sleepy and I need to finish this off plz plz plz

9. Sep 21, 2006

### StatusX

Hold on man. I can't help you if I don't understand you. What does n_2 mean? If it's $n^2$, check your algebra again (or make sure you copied the function right in the first post).

10. Sep 21, 2006

### matt grime

the symbol _ means subscript. ^ means superscript. and you have a mistake. StatusX is correct when he subs in (n^-2,n^-1), and you are not.

11. Sep 21, 2006

### precondition

ohhhhhhhhhhhhhhhhhhhhhhhhhhhhh
I copied down function wrong... it's supposed to be xy^2/(x^2+y^4!!!!)
my mistake sorry !!

12. Sep 21, 2006

### precondition

it's getting very late here and my brain is not functioning well.. sorry about that

13. Sep 21, 2006

### istevenson

suppose $$x = ky^2, y \rightarrow 0$$
then we know this: $$x \rightarrow 0$$ also.
so:
$$\lim \frac{xy^2}{x^2+y^4}= \lim \frac{ky^4}{k^2y^4+y^4}=\lim \frac{k}{k^2+1}$$
the limit doesnt exit, so its not continous at (0,0).