I have a question to this two dimensional function.
$f_1(x,y):=\begin{cases} \frac{2xy}{x^2+y^2},&\text{if }(x,y)\neq(0,0)\\0,&\text{else}\end{cases}$
I want to analyse if this function is continous for $(x,y)=0$
I think i can show this with an $\epsilon-\delta$ proof
$$|f(x,y)-f(0,0)|=\left|\frac{2xy}{x^2+y^2}\right|\leq 1$$
Would this be correct for the start? I dont know how to finish the proof.
Thanks.
Best Answer
Hint: A very frequently useful tool when the denominator is a relative of $x^2+y^2$ is to use polar coordinates. In our case we get that our function is $2\sin\theta\cos\theta$, which is highly dependent on $\theta$.
Alternately, we can observe that as $(x,y)\to(0,0)$ along the $x$-axis, the limit is $0$, while as $(x,y) \to (0,0)$ along the line $x=y$, the limit is $1$.
After we have discovered why the function is not continuous at $(0,0)$, we can if we wish write a formal $\epsilon$-$\delta$ proof that it is not continuous.