Stuck in assignment due tomorrow(in fact this morning)

[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..)
Any reply would be appreciated

 

[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.

 

[precondition]

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

hmm...

 

[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)

 

[StatusX]

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

 

[precondition]

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

 

[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?

 

[precondition]

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

 

[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).

 

[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.

 

[precondition]

ohhhhhhhhhhhhhhhhhhhhhhhhhhhhh
sorry my bad my bad apology*1000000000000000
I copied down function wrong... it's supposed to be xy^2/(x^2+y^4!)
my mistake sorry !

 

[precondition]

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

 

[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 doesn`t exit, so it`s not continuous at (0,0).