Consider the following theorem:
Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives $\frac{\partial f}{\partial x_j}$ exist on $F$ and are continuous at $x_0$, then $f$ is differentiable at $x_0$.
And I consider the converse of the above theorem:
Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If $f$ is differentiable at $x_0$, then all the partial derivatives $\frac{\partial f}{\partial x_j}$ exist on some neighbourhood of $x_0$ and are continuous at $x_0$.
It is trivial to show that the converse is NOT true when $m=1$. It seems no hope that it will be true when $m\geq 2$. Here is my question:
Is the converse true when $m\geq 2$? If is not true, how to construct the counterexample?
Edit: The title is corrected.
Edit: Since another question is not relevant to the first one here, I think, I put it into another post.