Wondering if anyone can point out a function $f$ from an open disc in $\mathbb R^2$ to $\mathbb R$, that:
- is defined and continuous at a point $p$;
- has(they exist) both of its partial derivatives at $p$;
- at least one of the partials is discontinuous at $p$;
- is NOT differentiable at $p$;
Context: (not necessary for providing an answer): While learning about the differentiablity of multivariable functions in my calculus course, started thinking about this particular case after finding out that the differentiablity of a continuous multivariable function (at a point) depends not solely on the existence of the partial derivatives. Naturally the question: “what other condition is necessary for differentiablity then?” arises. Learned that if both partials are continuous at p, that is sufficient. But apparently even if they aren't continuous at p, f can still be differentiable. Found one of those functions after much work but then wondered what would a function look like, that had everything set up for differentiablity at a point, but isn't, because of the discontinuity of one of the partials.
Would appreciate the function, or even a good intuition on how to find one.