I don’t think you can use the Intermediate Value Theorem to answer this. If taxonomists can entirely agree on one single path at each and every stage of evolution, the singular point of where an egg is now defined as a chicken egg where the egg that the creature which laid it hatched from is not a chicken egg–or vice versa, where a creature which is now defined as a chicken where its parents are not chickens–cannot be objectively determined. They’re human-defined lines, which makes this entirely a human philosophy problem in the first place.
(EDIT: messed up the formatting of this image) I like this analogy here:
It’s not completely relevant to this discussion, but it has some good points here. We can all agree that, at some point, it stopped being one color and started being another, but any method we use to draw that line would be arbitrary anyway. Maybe you take the hex code and find the point where the blue value is greater than the red value, but where is the text purple? Does purple even exist under this definition? Or maybe the text is red when, say, the hex for red is 80+% the total color value, blue for the opposite case, and purple for the in-between cases? But then, why 80% and not 90%? This is starting to sound really pretentious, but my point here is that in agreement to your last point, there’s no correct scientific answer to this problem.
If a tree falls in the forest and nobody is around to hear it, does it make a sound? Of course it does.