In my old book in calculus, it says that a sufficient condition for the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ to be differentiable at an internal point z, is that the partial derivatives exist at z and are continuous there.
Is this also a necessairy condition? Or is there a function defined in a domain areound z, where the function is differentiable at z, but there is no domain around z where the partial derivatives are continuous?
It is not a necessary condition, even for functions of one variable, as the following example shows: $$ f(x)=\begin{cases}x^2\,\sin\frac1x &x \ne0,\\ 0 & x=0. \end{cases} $$ $f$ is differentiable on $(-\infty,\infty)$, but $f'$ is discontinuous at $x=0$.