Define the function $f:\mathbb{R}^2\mapsto \mathbb{R}$ by \begin{equation*} f(x,y) := \begin{cases} \frac{\sqrt{|x||y|}}{|x|+|y|} & \textit{if} \; \; (x,y)\neq (0,0) \\ 0 & \textit{if} \; \; (x,y) = (0,0) \end{cases}. \end{equation*} Determine if $f$ continuous at $(0,0)$.
My attempt.
$f$ is continuous at $(0,0)$ if:
$f(0,0)$ exists
$\lim_{(x,y)\to (0,0)}f(x,y)$ exists
$\lim_{(x,y)\to (0,0)}f(x,y) = f(0,0)$.
If the limit is approached along the line $y = cx$, then \begin{equation*} f(x,y) = \frac{\sqrt{|x||y|}}{|x|+|y|} = \frac{\sqrt{|x|\times c|x|}}{|x|+c|x|} = \frac{\sqrt{c}|x|}{|x|(1+c)} = \frac{\sqrt{c}}{1+c}, \end{equation*} so $\lim_{(x,y)\to (0,0)} f(x,y) = \frac{\sqrt{c}}{1+c}$, which depends on $c$, thereby contradicting the uniqueness of limits which means the limit does not exist. Hence, $f$ is NOT continuous at $(0,0)$.
Is this correct? If not, any help would be great.
