I am looking for a topology on a set of functions which is somehow between uniform convergence and locally uniform convergence. For simplicity reasons I will explain my idea for the space of functions from $[0,\infty)$ to $\mathbb R$.
On this space, we can define the uniform topology: $f_n \to f$ uniformly if and only if $\sup_{t\in[0,\infty)} |f_n(t)−f(t)| \to 0$. We can also define the topology of locally uniform convegence: $f_n \to f$ locally uniformly if and only if for every $T>0$ it holds $\sup_{t\in[0,T]} |f_n(t)−f(t)| \to 0$ or equivalently $\sum_{T=1}^\infty 2^{−T} \sup_{t\in[0,T]} |f_n(t)−f(t)| \to 0$.
Consider now a sequence of numbers $t_n\in[0,\infty)$ converging to $t\in[0,\infty]$ (possibly infinity) and assume that $f(\infty)=\lim_{x\to\infty} f(x)\in\mathbb R\cup\{\pm\infty\}$ exists. Then $$f_n\to f\text{ uniformy} \implies f_n(t_n)\to f(t),$$ whereas we only have $$f_n\to f\text{ locally uniformy} \implies f_n(t_n)\to f(t)\text{ for all }t<\infty.$$ (I am not sure if these are even equivalent statements.)
In many cases, one actually observes something in between. Consider e.g. $f_n(t) = \min\{t,n\}$ and $f(t) = t$. Then, $f_n$ converges to $f$ locally uniformly but not uniformly. Nevertheless, for any sequence $t_n$ converging to $\infty$ slowly enough, i.e. $\frac{t_n}{n} \le 1$ for all $n$ large, we have $$f_n(t_n) \to f(\infty)=\infty.$$ But this fact cannot be deduced only from the local uniform convergence of $f_n$ to $f$.
So what I am looking for is a kind of semi-local uniform topology e.g. of the form $$f_n \to f \text{ if and only if } \sup_{t\in[0,n]} |f_n(t)−f(t)| \to 0.$$ Clearly, one could imagine to replace the interval $[0,n]$ by intervals of the form $[0,T_n]$ for any other sequence $T_n\to \infty$.
Is there any mode of convergence like this? I am also looking for any literature going in a similar direction.