Function continuous or not at (0,0)

[kottur]

Homework Statement

$f\rightarrowℝ$, $f(x,y)=\frac{x^{2}y}{x^{6}+y^{2}}$ where $(x,y)\neq(0,0)$ and $f(0,0)=0$.

Is the function continuous at $(0,0)$?

The Attempt at a Solution

I tried to find the limit at $(0,0)$ so I put $y=x$ into the function $f$ and got the limit 0 when $x\rightarrow0$. Tthen I put $y=x^{2}$ into $f$ and got the limit 1 when $x\rightarrow0$. That means that the limit does not exist right?
But the part that says $f(0,0)=0$ confuses me. Does that change the limit?

There is a second part for this problem where I'm supposed to find the first partial derivatives in $(0,0)$ or explain why they do not exist but I'd like to understand this first and then try to see if I can do the second part by myself. I think that if the limit does not exist in $(0,0)$ then the partial derivatives can not either by definition... But I'm not sure...

Thank you in advance.

 

[SammyS]

Homework Statement

$f\rightarrowℝ$, $f(x,y)=\frac{x^{2}y}{x^{6}+y^{2}}$ where $(x,y)\neq(0,0)$ and $f(0,0)=0$.

Is the function continuous at $(0,0)$?

The Attempt at a Solution

I tried to find the limit at $(0,0)$ so I put $y=x$ into the function $f$ and got the limit 0 when $x\rightarrow0$. Tthen I put $y=x^{2}$ into $f$ and got the limit 1 when $x\rightarrow0$. That means that the limit does not exist right?
But the part that says $f(0,0)=0$ confuses me. Does that change the limit?

There is a second part for this problem where I'm supposed to find the first partial derivatives in $(0,0)$ or explain why they do not exist but I'd like to understand this first and then try to see if I can do the second part by myself. I think that if the limit does not exist in $(0,0)$ then the partial derivatives can not either by definition... But I'm not sure...

Thank you in advance.

You asked:

"But the part that says $f(0,0)=0$ confuses me. Does that change the limit?"​

The answer is no!

The limit as (x,y) → (0.0) has nothing to do with the value of f(0,0). Indeed, this limit can exist even if f(0,0) is undefined.

By The Way: Your solution to this problem is correct. You have shown that the limit does not exist.

 

[kottur]

Thank you.