Let $A \subseteq \mathbb{R}^n$ be closed with no isolated points and $f:A \to \mathbb{R}^m$. Suppose that for every point $x_0 \in A$ we have (at least one) matrix $L_{x_0}$ such that $$ \lim_{x,y \to x_0} \frac{f(x)-f(y) - L_{x_0}(x-y)}{\Vert x-y\Vert} = 0 $$ The existence of this limit is always possible because $x_0$ is no isolated point of $A$.
My question is: Is there an open set $U\subseteq \mathbb{R}^n$ with $A \subseteq U$ and a differentiable function $g$ on $U$ such that $g\vert_A = f$? What if we say that $L_{x_0}$ is unique for every $x_0 \in A$? What if we say that $x_0 \mapsto L_{x_0}$ is conitinuous?
In a book about differentiable manifolds, I read that a function is called differentiable on a border point if it can be continued differenitably on a neighbourhood of that point. I was wondering what's the point. Why don't we use pointwise differentiability like usual?