In my book, I found a problem, asked to check the differentiability of the function $f(x,y)$ at $(0,0)$ where the function f$(x,y)$ is defined as follows
$$ f(x,y) = \begin{cases} \large\frac{xy}{\sqrt {x^2+y^2}}, & \text{if $x^2+y^2 \neq (0,0)$} \\ 0, & \text{if $x^2+y^2 = (0,0)$} \end{cases} $$
Now in the book, they give a solution using alternative definition of differentiability at a point $(0,0)$ and there they have approached to a contradiction that $\frac{1}{2} = 0$
So, they proved that the function is not differentiable at all.
But when I tried to solve the problem by proving that the $f_x$ and $f_y$ exist at $(0,0)$ and the function is continuous at that point too. So basically, I prove that the given function is differentiable at that very point.
But the given solution in my book makes me confuse. I don't understand what actually happen, Did I do something wrong if I did please make me clear? Because at the same time these two solution is not possible for this function.
And another thing can I use "sufficient condition for differentiability" for this function?
Pardon me if I did any wrong.
Thank you.