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Suppose we have a map $f: S \rightarrow \mathbb{R}^{n}$, where $S \subset \mathbb{R}^{m}$, such that for each $a \in S$ there exists an $m$ by $n$ matrix $A$ such that

$\lim_{h \rightarrow 0}\frac{f(a+h)-f(a)-Ah}{|h|} = 0.$

What conditions must be satisfied so that $f$ can be extended to a differentiable function defined on an open set containing $S$? I know that if $f$ can be locally extended to a differentiable function, then $f$ can be extended in the desired way. However, is there a more general result...possibly an if and only if condition?

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If $S$ is closed, and $A$ is suitably continuous, you can apply Whitney extension theorem to extend $f$ to be real analytic outside $S$.

If $S$ is not closed, then in a neighborhood of $\bar{S}\setminus S$ you can easily construct an example of a function that is $C^\infty$ on $S$ but cannot be continuously extended to $\bar{S}$ (think $1/|x|$ on the punctured disk).

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For various versions of Whitney type extension theorems, a decent reference is E. Stein's Singular Integrals and Differentiability Properties of Functions, Chapter VI. – Willie Wong Jul 13 '11 at 16:23
+1 because of the great book reference. – Jonas Teuwen Jul 13 '11 at 17:29

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