While studying rectification of curves, I considered a curve and to measure its length in a different fashion, and arrived at a problem. I would like to clarify the confusion in my understanding.
Consider a unit square in plane with vertices (0,0), (0,1),(1,0), (1,1). The diagonal joining (1,0) and (0,1) has length $\sqrt{2}$, well known. Suppose I approach theis diagonal in the following way: first by the path $P_1: (1,0)-(1/2,0)-(1/2,1/2)-(0,1/2)-(1,0)$. This is like tow "L"s, with top end of one joined to bottom end of other- forming stairs. The length of this path is $2$. Next, we form path $P_2$ with four "L"'s, in a nice way to form stairs. Again the length of this path is $2$.
We see that the sequence $\{P_n\}$ of paths, which are piecewise differentiable functions (?), converges to the diagonal $(1,0)-(0,1)$. But the length of each path is $2$, but we cant conclude that the diagonal should have length $2$. Why such a contradiction arises?
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