Say we have a convergent sequence $(x_n)$ where $x_n \in E$ for all $n \in \mathbb{N}$ and $E$ is a subset of a metric space $(X,d)$.
With this setup, we usually define it's limit as a point $x \in X$ such that for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $n > N$ implies $d(x_n,x) < \epsilon$.
In some sense, I think that the above definition of the limit effectively mean that given any "$\epsilon > 0$, there is some kind of $N$ that we can use to get the sequence within $\epsilon$ of the limit $x$.
I am wondering whether this can be reformulated as follows: there is a function $f: \mathbb{R} \rightarrow \mathbb{N}$, so that $f(\epsilon) = N$ and $n > N$ implies $d(x_n,x) < \epsilon$. If so, the function $f$ would have some nice properties (it would be onto, and monotonically decreasing in $\epsilon$ for instance).
Is there any use to thinking about functions in this way / has it been introduced in this way?