Prove that if all the partial derivatives of $f: \mathbb{R}^{p} \rightarrow \mathbb{R}$ (with $p \geqslant 2$ ) exist in an open set containing $(0,0, \ldots, 0)$ and are continuous at this point, then $f$ is differentiable at $(0,0, \ldots, 0)$.

Let

$$g(x)= \begin{cases}x^{2} \sin (1 / x), & x \neq 0 \\ 0, & x=0\end{cases}$$

and

$$f(x, y)=g(x)+g(y) .$$

At which points of the plane is the partial derivative $f_{x}$ continuous?

At which points is the function $f(x, y)$ differentiable? Justify your answers.