Showing that the function given by $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ and $f(0,0)=0$ is continuous but not differentiable 
Let
$$
 f(x,y) =
\begin{cases}
\dfrac{xy}{\sqrt{x^2+y^2}}   & \text{if $(x,y)\neq(0,0)$ } \\[2ex]
0 & \text{if $(x,y)=(0,0)$ }  \\
\end{cases}
$$
Show that this function is continuous but not differentiable at $(0,0),$ although it has both partial derivatives existing there.


I can show this function is continous and the partial derivatives exist. But how can I show that this function is not differentiable?
Is showing that the function is differentiable similar to showing that a derivative exists?

Since $|f|\le|x|$, $\lim_{(x,\,y)\to(0,\,0)}f=0$. But $\frac{\partial f}{\partial x}=\frac{y^3}{(x^2+y^2)^{3/2}}$, which $\to1$ along an $x=0,\,y\to0$ path, or $\to0$ along an $x\to0,\,y=0$ path.
