The problem I have: For each $n \in \mathbb N$ I have
$$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ D_n & = \{ x_k^n : 0 \le k \le n \land k \in \mathbb N\} \\ f_n & : D_n \rightarrow \mathbb R^{+} \\ \Delta f_n & : D_n\setminus\{x_n^n\} \rightarrow \mathbb R: x_k^n \mapsto \frac{f_n\left(x_{k+1}^n\right)-f_n\left(x_k^n\right)}{h_n} \end{align} $$
with
$$\begin{align} \Delta f_n\left(x_k^n\right) & = g\left(x_k, f_n\left(x_k^n\right)\right) \text { for a function } g : \mathbb R^2 \rightarrow \mathbb R\\ \sum_{k=0}^n f_n\left(x_k^n\right) \cdot h_n & = 1 \\ \lim_{n\rightarrow\infty} h_n & = 0 \\ \lim_{n\rightarrow\infty}x_0^n & = -\infty \\ \lim_{n\rightarrow\infty}x_n^n & = \infty \end{align}$$
I need to find a function $f:\mathbb R\rightarrow \mathbb R$ which approximate $f_n$ (if it exists), i.e. the function $f:\mathbb R \rightarrow \mathbb R$ shall satisfy $$\lim_{n\rightarrow\infty} \max\left\{\left|f\left(x_k^n\right)-f_n\left(x_k^n\right)\right|: 0 \le k \le n \land k \in \mathbb N\right\} = 0$$
My question: Is there any work already have discussed the above problem? Are there any useful theorems for my problem? What kind of literature should I read?
I guess reading about numerical solutions of ODEs will be useful for me but I thought asking this question here will give me some additional hints...
My attempt to solve the problem: Because of the two properties $$\begin{align} \Delta f_n\left(x_k^n\right) & = g\left(x_k, f_n\left(x_k^n\right)\right) \text { for a function } g : \mathbb R^2 \rightarrow \mathbb R\\ \sum_{k=0}^n f_n\left(x_k^n\right) \cdot h_n & = 1 \end{align}$$ I would take the function $f:\mathbb R\rightarrow\mathbb R$ with $$\begin{align} f^\prime(x) & = g(x,f(x)) \\ \int_{-\infty}^\infty f(x) &= 1 \end{align}$$ Of course it's left to investigate under which circumstances $f$ is the right solution...