In this question it is asked what the upper-bound of the ratio of a solid object's surface area can be visible through direct, unaided observation.
The accepted answer says that there is no such upper-bound, referring to a function whose mathematical limit is $1$ as the value $r \rightarrow 0$.
I am not sure how to interpret this. I see two interpretations; one that is correct but perhaps somewhat misleading* $(1)$, and one that is just plain illogical $(2)$.
$(1)$ There is no upper-bound, because the proportion grows arbitrarily close to $1$ and due to the density of the reals, there is no one number to grasp at. This is true.
$(2)$ The upper-bound is $1$, because the function is, mathematically, continuous over $r=0$. Its analogical description of physical reality however, ends at $r=0$. As such, viewing the function as a mathematical model of a physical transformation, the function is discontinuous at $r=0$, as an a priori truth following from the definition of a cone (it obviously can't be a cone if its radius is $0$).
If the answerer's intended interpretation was $(2)$, then this seems to me as a situation of someone conflating a mathematical analogy with the physical reality it is describing. Just because the math say that $r$ can be $0$, doesn't mean that physical reality allows $r=0$.
*The answer itself isn't really misleading, but the answerer's comments in the discussion confused me. John Deaton pointed to the fact that the answer did not explicitly rule out $r=0$ as a possibility. The answerer said that "the answer left that possibility untouched", and that they were "uninterested in that analysis". This means that despite the fact that $(1)$ is the only logical interpretation for the cone, the answerer explicitly stated that both $(1)$ and $(2)$ are possible interpretations. Their language (to me) within the answer seems to suggest $(1)$, but their comments in the discussion states either is possible. The most confusing part is that I find $r=0$ to be trivially illogical; but the answerer stating they were "uninterested in that analysis" suggest that from their point of view, it isn't trivial, but rather something to be analyzed. Given that they appear quite competent to me, this makes me doubt my feeling that $r=0$ is trivially illogical.
The question:
If $r=0$, there is no base, and thus, there is no cone. Given that this is a simple fact, why is the elimination of the case of $r=0$ referred to as an analysis, when it appears to be a fact trivially establishable? Is this as simple as I think it is, or is there more going on here?
