I am wondering if the following statement is a canonical theorem in real analysis. Does anybody here know the exact reference for it? It maybe a corollary of the mean value theorem, I think. Motivated by the answer to this question, I curious about that if this statement is still true in the higher dimension case.
The following statement is a corollary of the Mean Value Theorem.
Let $x_0\in{\bf R}$ and $U(x_0)$ be the neighborhood of $x_0$.
$f:U(x_0)\to{\bf R}$
is continuous on $U(x_0)$ and differentiable on $U(x_0)\setminus\{x_0\}$. If the limit
$\lim_{x\to x_0;x\in U(x_0)\setminus\{x_0\}}f'(x)$
exists, then $f$ is differentiable at $x_0$ and
$f'(x_0)=\lim_{x\to x_0;x\in U(x_0)\setminus\{x_0\}}f'(x)$
Here is my question:
Is this statement still true for the higher dimension if $f:U(x_0)\subset{\mathbb R}^n\to{\mathbb R}^m ?$
Since I don't know any generalization of the MVT in the higher dimension, I think one may need to approach it directly by definition. On the other hand, if one can deal with it "component-wise", one may be able to reduce it to the one dimension case.