I am trying to get a function $f:\mathbb{R}^2 \to \mathbb{R}$ that is differentiable at the origin but discontinuous everywhere else?
As a simpler case, we have that $$g\left(x\right)=\begin{cases} x^2 & \mbox{if }x \in\mathbb{Q}\\ -x^2 & \mbox{otherwise} \end{cases}$$ has this property. Can we use this to help us construct an $f$?