I am looking for a proof for the Jordan curve theorem in the particular case of rectifiable curves. I consider this case as "intermediate" between the general case and the case of piecewise differentiable (or $C^n$) curves, so I think that, possibly, a proof with "intermediate" difficulty must exist. (A reference for the proof in the case of differentiable curves will be welcomed.)
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1This http://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem. – Takahiro Waki Apr 03 '17 at 17:17
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Any reason you think the rectifiable case is simpler than the general case? The smooth case is indeed easier (but still uses Sard's theorem, which is not completely elementary). – Moishe Kohan Apr 03 '17 at 23:23
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@MoisheCohen I think that the fact the curve is rectifiable implies that the polygonal approximations are "better" behaved... but anyway, this is just a gut feeling. – Matemáticos Chibchas Apr 03 '17 at 23:39
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1Note that polygonal approximations may (and frequently will) be by non-simple polygonal curves. I do not see how these would help you. – Moishe Kohan Apr 04 '17 at 00:04
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@MoisheCohen You are right, the hypothesis seems to be of no help. – Matemáticos Chibchas Apr 05 '17 at 16:52