Prove that$$\begin{array}\\&f(x)&=\begin{cases}\frac{x|y|}{\sqrt{x^2+y^2}}&\text{ if }(x,y) \ne (0,0)\\0&\text{ otherwise }\end{cases}\end{array}$$ is differentiable in $(0,0)$.
Proof: We know that $f$ is differentiable in $(0,0)$, if all partial derivatives exist and are continuous in $(0,0)$. We have $$\frac{\partial f}{\partial x}=\lim_{h\to\ 0}\frac{f(h,0)}{h}=0$$ $$\frac{\partial f}{\partial y}=\lim_{h\to\ 0}\frac{f(0,h)}{h}=0$$ So the partial derivatives exist.
I am now stuck at proving the continuity of the partial derivatives in $(0,0)$. Can anyone help me with that?