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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.

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    One last comment, a complement to this problem: Let $x_0\in {\mathbb R}$ and $U(x_0)$ be the neighborhood of $x_0$. $f:U(x_0) \rightarrow {\mathbb R}$. is continuous on $U(x_0)$ and differentiable on $U(x_0)\backslash \{x_0\}$. If the limit $\lim_{x \to x_0}f'(x)$ doesn't exist, $f$ can still be differentiable at $x_0$.2011-05-21

3 Answers 3