Does there exist a function $f:\mathbb{R}^n\to \mathbb{R}^m$ with the property:
$\exists x_0\in\mathbb{R}^n$ such that $\frac{\partial f_i}{\partial x_j}(x_0)$ exists for all $i,j$, but $f$ is discontinuous at $x_0$.
I konw that all partial derivative exist does not imply total differentiability. But what about continuity? I am curious about this.