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 √2, well known. Suppose I approach theis diagonal in the following way: first by the path P1:(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 P2 with four "L"'s, in a nice way to form stairs. Again the length of this path is 2.
We see that the sequence {Pn} 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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