real analysis - Rectifiable Curve - Mathematics Stack Exchange The curve is not rectifiable if we can find a "zig-zag" approximation for the curve with arbitrarily large arc length Pick n ∈N n ∈ N greater than 3 3, and for each 0 <i <n 0 <i <n, let ti = 1 i t i = 1 i, and let tn = 0 t n = 0, t0 = 1 t 0 = 1
Rectifiable sets and examples - Mathematics Stack Exchange A countable union of rectifiable sets need not be rectifiable For example, a singleton {r} {r} is rectifiable, but a countable set, dense in the unit ball, is not rectifiable, although it is a countable union of those singletons
Rectifiability of a curve - Mathematics Stack Exchange It follows that γN γ N has a total length > 2 π ∑N−1 k=2 1 k> 2 π ∑ k = 2 N − 1 1 k Since the sum is unbounded for N → ∞ N → ∞ the considered curve γ γ is not rectifiable
A function with rectifiable graph satisfies Lipschitz condition on a . . . If we merely assume that the graph is a rectifiable set, then there is a counterexample Let E be a fat Cantor set and f =χE The graph is contained in the union of two lines, and the Lipschitz condition fails on any set of sufficiently large measure