Consider a discrete system of particles. For each particle, you can assign a position vector $$ \vec{r}_n(t) = f_n(t)\hat{x} + g_n(t)\hat{y} $$
There will be $N$ position vectors for the $N$ discrete particles. Now consider an electric field which varies in space and time
$$ \vec{E}(x,y,z,t)$$
Space is a continuous quantity. What I don't understand is why there is only a single function to describe a quantity which varies over space. For instance, why don't we have an infinite and uncountable number of equations for the electric field due to the infinite and uncountable places in space? For ease, I will consider a single dimension of space. Why is it that
$$ \vec{E}_{x_1}(t), \vec{E}_{x_1 + dx}(t), \vec{E}_{x_1 + 2dx}(t) \tag{1}$$ is not the form of the electric field? Instead, it would take the form $$ \vec{E}(x,t)\tag{2}$$
I ask because equation $(2)$ is much more restrictive that equation $(1)$. Equation $(2)$ is one equation, whereas equation $(1)$ is a set of equations which can take wildly different forms between an $x$ and an $x + dx$ location.
Question rephrased in a slightly different way: Why is that when a label $n$ goes from discrete to continuous, the label is placed within the function $f(n, )$. If the label is discrete, it is place as a subscript $f_n( )$.
Thoughts: $\vec{E}(x,t)$ is restrictive. A single form for all $x$. Do we restrict it like this because we assume something about continuity?