Determine continuity of two variable function

Question

Examine the following function for continuity

$f(x,y)= \begin{cases} \frac{y}{|y|}\sqrt{x^2+y^2}, & \text{if $y\neq0$} \\[2ex] 0, & \text{if $y=0$} \end{cases}$

i think function is not continuous for any point $(x,0),x\neq0$ . this point can be approach through vertical line passing through in upper half plane that point giving me limit $|x |$, which is not equal function value. At any other point function is continuous. right??

Answer 1

Let $(x_0,0) \in \mathbb{R}^2$ and let $x_n = (x_0,1/n)$ and $y_n=(x_0, -1/n)$. Then $f(x_n) = \sqrt{x_0^2 + 1/n^2} \longrightarrow |x_0|$ while $f(y_n) = -\sqrt{x_0^2 + 1/n^2} \longrightarrow - |x_0|$. Therefore if $x_0 \neq 0$, $f$ is not continuous at $(x_0,0)$. It's left to check what happens when $x_0=0$, though it's quite clear that $f$ is continuous at $(0,0)$