Prove a function is not continuous

[ssayan3]

Homework Statement

Using the epsilon-delta definition, prove that the function f:R$$^{2}$$ $$\rightarrow$$ R by f(x,y) = xy/((x$$^{2}$$) + (y$$^{2}$$)), and f(0,0) = 0 is not continuous.

The Attempt at a Solution

I just really have no clue how to set up a delta-epsilon proof for functions that involve quotients. I went ahead and set up as much delta information as I could, but I have no idea how to set up the epsilon part:

|x-x$$_{0}$$|<$$\delta$$, |y-y$$_{0}$$|<$$\delta$$, and |(x-x$$_{0}$$)+(y-y$$_{0}$$)|<$$\delta$$ (Those are supposed to be x (initial) and y (initial) for the delta info... I couldn't get LaTex to set them up correctly...)

Can someone give me a couple of good pushes in the right direction? :-)

 

[HallsofIvy]

For problems like these, I recommend changing to polar coordinates. That way, (x,y)-> (0,0) becomes just r-> 0 no matter what $\theta$ is. To show that this function is not continuous you only have to show that $|f(r, \theta)|$ cannot be made arbitrarily small ("$< \epsilon$") for small r for at least some values of $\theta$. For this particuar problem I think you will find that easy.

 

[ssayan3]

Hm, I'm just not sure why I didn't think of that before! Thank you!