I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following.
Definition (Blob) Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The Blob of $S$ at $p$ is defined to be the subset $$ \left\{v \in \mathbb{S}^{n-1} \middle| \exists (x_n) \subset S, x_n \to p, \frac{x_n - p}{\|x_n - p\|} \to v \right\} $$
Definition (Widget-differentiable) Let $S\subset \mathbb{R}^n$ be a set, and $p\in S$. Let $v$ be in the Blob of $S$ at $p$. We say that a function $f:S \to\mathbb{R}$ is Widget-differentiable in the direction $v$ with derivative $\eta$ if for all sequences $(x_n)\subset S$ with $x_n \to p$ and $\frac{x_n - p}{\|x_n - p\|} \to v$, we have the limit $$ \lim_{n \to \infty} \frac{f(x_n) - f(p)}{\|x_n - p\|} = \eta.$$
The above is, of course, an attempt to generalise the notion of differentiability and derivatives to functions defined on subsets of $\mathbb{R}^n$, which are not necessarily manifolds, but which have accumulation points. In particular, if $M\subset\mathbb{R}^n$ is a submanifold and $M \supset S$, then the Blob of $S$ is a subset of the unit-sphere bundle on $M$. And if a function $f$ on $M$ is differentiable at $p$, its restriction to $S$ is Widget-differentiable at $p$.
My Question:
What are the actually names of "Blob" and "Widget"? Does anyone remember reading the same paper that I read which described this generalisation?
