This relates to a previous question I asked, so we can somewhat pick up from there.

For the function $\displaystyle f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right$

determine at which points $\displaystyle f$ is differentiable.

I already know that $\displaystyle f$ is continuous ONLY at the point $\displaystyle (0,0)$, so that's the only potential location for where $\displaystyle f$ could be differentiable. Showing whether or not it is, however, is the trick.

Would using this trick work?

$\displaystyle u(h,k)=\frac{f(x+h,y+k)-f(x,y)-f_x(x,y)h-f_y(x,y)k}{\sqrt{h^2+k^2}}$ when $\displaystyle (h,k)\ne (0,0)$

I only need to determine whether or not $\displaystyle u(h,k)\rightarrow 0$ as $\displaystyle (h,k)\rightarrow (0,0)$ to prove whether or not the function is differentiable at $\displaystyle (0,0)$.