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? 



Generalisation of "tangent space" to not-necessarily connected sets
Ah, I think I found the answer. (For future reference these notions apparently come up in *non-smooth analysis*.) The "Blob" I defined is closely (and rather trivially) related to the notion of the **contingent cone** or **Bouligand tangent cones** (the [French Wikipedia has a better article than the English on the subject](http://fr.wikipedia.org/wiki/C%C3%B4ne_tangent)). 


