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?

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    $\begingroup$ Why do you expect there to be continuous behavior at a singularity? $\endgroup$ – G. Smith Oct 27 '19 at 23:42
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    $\begingroup$ When you look at the scale factor rather than the “size”, nothing goes from 0 to $\infty$ in an instant. $\endgroup$ – G. Smith Oct 27 '19 at 23:44
  • $\begingroup$ We don't really know whether the universe has infinite size or not, so what bothers you did not necessarily happen. $\endgroup$ – G. Smith Oct 27 '19 at 23:45
  • $\begingroup$ In the first paragraph, you say the domain is $(0,\infty)$, which makes sense. But then in the second paragraph, you start talking about a domain of $[0,\infty)$. Why? $\endgroup$ – user4552 Oct 28 '19 at 1:27
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/9419/2451 and links therein. $\endgroup$ – Qmechanic Oct 28 '19 at 4:31