TITLE: Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
QUESTION [7 upvotes]: How would you prove or disprove that the function given by
$$f(x,y) = \begin{cases} \dfrac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$
is continuous at $(0,0$)?

REPLY [8 votes]: Observe that 
$$
\left| \frac{x^2y}{x^2+y^2} \right| \leq \frac{x^2 |y|}{x^2} = |y|
$$
provided $x \neq 0$ and $y \neq 0$. Then you conclude.