The are several notions of sizes with respect to the universe which are defined by some type of horizon. As I understand, these are all finite, and their change is continuous over time. So for any of these notions there exists a continuous function $f : \mathbb{R}^{+}_0 \rightarrow \mathbb{R}^{+}_0$ that maps a point in time after the big bang to size of the universe at that point in time.
Then there is another notion, the total size of the universe, and as e.g. wikipedia states, it is consistent with theory to be infinite in size. It is obvious that a function with a codomain of $\mathbb{R}^{+}_0$ cannot be an appropriate model for an infinite size. And I have trouble to even categorize a proper candidate. If the universe is infinite now, and its expansion is continuous, then it was infinite yesterday, a year ago, 13 billion years ago. Which kind of function goes from $f(0) = 0$ to $f(t) = \infty, t > t_0$ for some $t_0 > 0$, and how does it do that continuously?
