Showing continuity of $\dfrac{xy(x-2y)}{x^2+y^2}$ at the origin. Show that the function $f(x, y)=\dfrac{xy(x-2y)}{x^2+y^2}$ is continuous on $\mathbb{R}^2$.


It is easy to show continuity of $f(x, y)$ at points other than the origin but how is it continuous at the origin?
Hint: $$\Big|\dfrac{xy(x-2y)}{x^2+y^2}\Big|=\Big|\dfrac{x^2y}{x^2+y^2}-\dfrac{2xy^2}{x^2+y^2}\Big|\leq|y|+2|x|$$
The function $f(x,y)$ is undefined at the origin, hence, it is not continuous at $(0,0)$. 

Note that it does not have a limit as $(x,y)\to (0,0)$ because the limit of $f$  is different depending on the path that $(x,y)\to (0,0)$. For example,

$$ \lim_{y\to0} f(0,y) = -2$$
$$ \lim_{x\to0} f(x,0) = 1$$