Here is an old Berkeley Preliminary Exam question (Spring 79).
Let $f : \mathbb{R}^n-\{0\} \rightarrow \mathbb{R}$ be differentiable. Suppose
$\lim_{x\rightarrow0}\frac{\partial f}{\partial x_j}(x)$ exists for each $j=1, \cdots ,n$.
(1) Can $f$ be extended to a continuous map from $\mathbb{R}^n$ to $\mathbb{R}$?
(2) Assuming continuity at the origin, is $f$ differentiable from $\mathbb{R}^n$ to $\mathbb{R}$?
End of question.
In the book by De Souza, the following solution is given for (1)
No, with the counter example $f(x,y)=\frac{xy}{x^2+y^2}$ for $(x,y)\neq (0,0)$.
This is not an extendable function, but $\lim_{x\rightarrow0}\frac{\partial f}{\partial x_j}(x)$ does not exists. I think the solution is wrong, any other correct counter example?
Thanks