Given a function $f(x,y) = \frac{y^2}{x}$, defined as $0$ at $x=0$, I want to show that this function is not continuous at $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.
(I guess my zeroth question is: is this true?)
My argument is this: Let$ \begin{pmatrix} x \\ y \end{pmatrix}$ go to $0$ along the path $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} t \\ t^{1/2} \end{pmatrix}$. This path converges to $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$but $f(x,y) = 1$ along this path for $x \neq 0$.
Is the gist of this argument correct?