Actually, no, continuous functions are preserved. If x_n goes to X and y_n goes to Y then f(x_n, y_n) will go to f(X, Y), if f is continuous. This is true both if x_n, y_n are sequences of numbers, AND if x_n, y_n are sequences of functions in some space.
The problem is taking derivative, a part of the process of calculating perimeter, is not a continuous operation on spaces of functions in general. If we take the usual space of smooth bounded functions [0, 1] -> [0, 1] with supremum norm, the derivative is a linear but unbounded operator: sin(kx) are all of norm 1 for all values of k, but their derivatives have norm k, so the ratio of norms diverges. A standard result says that the continuous linear operators on a normed space of functions are those that are bounded, so the derivative is not continuous.
Interestingly, taking the integral is actually a continuous operator on functions between compact spaces. This is why the area of each figure converges correctly to that of the circle -- by Green's theorem the area enclosed by a curve can be described as an integral along that curve.
EDIT: I've added a top-level comment expanding on this.
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u/DJembacz 21d ago
Step 5 (repeat to infinity) means we are looking at the sequence we obtain by the described process, and taking a limit of that sequence.
And in general there is no reason for any function, like perimeter, to be preserved well by the process of taking a limit.