Convergence of functions with different domain Question: Is there a concept of convergence for functions $f_n: D_n \rightarrow X$ with different domains to a function $f: D \rightarrow X$?
I know concepts like uniform convergence or almost everywhere convergence but they require that all functions $f_n$ and their limit $f$ share the same domain...
Example: Take $D_n = \{ \tfrac kn : 0 \le k \le n, k \in \mathbb N \}$. Let $f_n$ be defined via
$$f_n : D_n \rightarrow \mathbb R: \tfrac kn \rightarrow \tfrac kn + \tfrac 1n$$
The limit I have in mind would be $f: [0,1]\rightarrow \mathbb R: x \mapsto x$. 
I guess, if one requires from the limit to be continuous, the above limit $f$ is unique. But I'm not sure, because I do not know which concept of limit I can apply...
 A: In theory of stochastic processes, one can characterize a process by a linear operator $\mathcal A$ acting on the continuous functions defined over the state space of the process. Such operator may not be defined for any function, so in fact it comes with its own domain of definition and strictly speaking we need to talk about a pair $(\mathcal A,D)$. Now, when it comes to the convergence of stochastic processes in distribution, we can show that by showing that $\mathcal A_n\to \mathcal A$ and $D_n\to D$. I guess a similar procedure may be of use in semigroup theory in general.
See for example Theorem 4.1 here: there in condition $(4.3)$ we require that any element of a limiting domain $x\in D$ must be a limit of a sequence $x_n\in D_n$.
A similar approach can be used in your example: let's say that $(f_n,D_n) \to (f,D)$ if

for any $x\in D = [0,1]$ there exists a sequence $x_n\in D_n$ s.t. $x_n\to x$ and $f_n(x_n)\to f(x)$

which is indeed the case. Now, the question would be: how useful is such definition of limits in applications we are bearing in mind when coming up with it.
