Determine where the Functions are Continous $$f(x,y)=\begin{cases}\dfrac{xy}{x^2+y^2}&(x,y)\neq(0,0)\\
0 &(x,y)=(0,0)\end{cases} $$
I use graph tool to help me visualize this problem, and I think it is continuous for all the graph, but the answer is wrong. Can someone guide me how to solve this kind of question? 
 A: You should try to find the limit for this function when both $x$ and $y$ approach $0$, then compare the limit to see is it $0$ or not. If the limit $0$, then the function is continuous everywhere. However, if it's not, then the function is discontinued at $(0,0)$.
Hint: In this problem, you can use $y=mx$ to find the limit and see the result. The graph may be continuous, but it may be the limit is close to $0$ and display as a continuous graph. 
A: Elementary properties of continuous functions should allow you to convince yourself that $f$ is continuous away from the origin. You should worry about the origin. $f$ is continuous at the origin iff $\lim_{(x,y) \to (0,0)} f(x,y) = f(0,0) = 0$. If $\lim_{(x,y) \to (0,0)} f(x,y)$ exists, we should be able to take it along any path towards the origin and all such paths should give the same answer. Taking the limit along $y=x$ and taking the limit as $x \to 0$ gives $\lim_{x\to 0} f(x,x) = \frac{x^2}{x^2 + x^2} = \frac{1}{2} \neq 0$. This shows that $\lim_{(x,y) \to (0,0)} f(x,y)$ can  not possibly $=0$, hence $f$ is not continuous at the origin.
A: It is continuous everywhere except the origin. At origin take  $y=mx$ then
$\lim_{x\rightarrow 0, y\rightarrow 0} \frac{xy}{x^2+y^2}=\frac{m}{1+m^2}$ the value of limit depends on 'm' and hence limit does not exist at the origin.
