TITLE: Can we speak of derivatives of sets?
QUESTION [11 upvotes]: Suppose we have a monotone sequence of sets: $$A_1,\ldots,A_n$$
$$A_i \subseteq A_{i+1}$$
I think this is a function from $\mathbb N$ to a space of sets. Can we define a function from $\mathbb R$ to a space of sets? Could we then define a derivative of this function?

REPLY [3 votes]: To begin with, there is no "space of sets". Even a "set of all sets" does not exist (in ZFC and its extensions, for example). Your functions should have values in some set, whose elements are sets themselves.
$\mathbb{N}$ and $\mathbb{R}$ are sets also, and nothing prevents you from speaking of a function from them to somewhere with values being sets. For example $f: \mathbb{R} \to 2^\mathbb{R}$, $f(x) = \{y : y < x\}$.
To talk about derivatives, you have to define a structure of a Banach space on the function's codomain (that is, roughly, you have to define what do addition, multiplications by scalar, size and limits mean for your sets). If you do this, then you can define a derivative.