Condition when the sum of small lengths of limiting curve is equal to limit of sum of small lengths of curve

Question

The staircase paradox, or why $\pi\ne4$ in this thread i understand whats the main reason the argument fails ,

(if i understand correctly it all boils down to the fact that "Limit of sequence of curves length is not equal to sum of lengths of limit of the curve") .

So my question is when do they both actually are equal ? I mean is it similar to what we know in basic limits that

Limit of sum of functions is equal to sum of limit of functions provided sum of individual limits exist ?

So maybe continuity is best for such functions? In the intial linked problem sequence of curves the path length function maybe is discontinuous at some points thats why its not being equal ? something else is the condition when both are equal ?
That is maybe individual length limit must exists of all portions of curve ?

Some of the references in my comments to this similar question asked 5 days ago deal with your questions. — Dave L. Renfro, Jul 19 '22 at 09:26
For curve $(x(t),y(t))$ one can write $l=\int_{t_0}^{t_1} \sqrt{x'(t)^2+y'(t)^2}, dt$. I believe, if two "almost the same" curves are such that $x_1(t)-x_2(t)$ and $y_1(t)-y_2(t)$ and $\sqrt{x_1'(t)^2+y_1'(t)^2}-\sqrt{x_2'(t)^2+y_2'(t)^2}$ are infinitely small then they will have the same length at any part of them. — Ivan Kaznacheyeu, Jul 19 '22 at 10:06

Condition when the sum of small lengths of limiting curve is equal to limit of sum of small lengths of curve

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