A definition of function could be the following:
A function is a tuple $(f, C, D)$, where $f\subseteq C\times D$ satisfying:
- $\forall x\in C : \exists y \in D : (x, y) \in f.$
- $\forall (a,b), (c,d)\in f : a= c \rightarrow b = d$.
Notation: If $(x,y)\in f$, given that such $y$ is unique, we write $y=f(x)$.
In physics it's useful to have the notion of so-called "functions of position": The force of gravity exerted by the Earth is a good example, say, if you're in one dimension (taking masses and constants to be $1$), this force could be:
\begin{align} F:\ &\Bbb R^+ \to \Bbb R\\ &k\mapsto \frac 1 {k^2} \end{align}
One usually then writes the equation of motion for a particle $(m = 1, x: \Bbb R\to \Bbb R)$
$$F= \ddot x\tag{1}$$
Notice how, in the definition of force, which is an absolutely correct definition for a function, nowhere it is required that the input is a so called 'position' argument. So if this additional, informal information wasn't given, one could not go from $(1)$ to
\begin{align} \frac{1}{x(t)^2}=\ddot x(t) \end{align}
To actually have a differential equation to solve.
See the problem here?
Perhaps I'm just missing something obvious, but saying something like "gravity is not a function of time" doesn't make sense from the definition of a function: it doesn't matter what you call your variable.
I guess the question is, how do we mathematically represent these physical scenarios in an accurate, not handwavy way?