# Is it continuous at $(0,0)$?

$$f(x,y)=\begin{cases} \frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0 \\ 0, \text{ if } x^2+y^2=0 \end{cases}$$

Is it continuous at $(0,0)$?

-

No, since along line $y=x$ $$\lim_{x,y\to0}f(x,y)=\frac12\neq0=f(0,0).$$
No, to disprove show sequential continuity doesn't hold. Look along the sequences $\{ x = 1/n , y = 1/n \}$ and $\{ x = 0 , y = 0 \}$ (the constant sequence)