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The non-differentiability is the non-existence of this limit The limit being, in any case, physically unattainable (in nite energy is required to reach it, according to Heisenberg time-energy relation), v is rede ned as v(t, dt), function of time t and of the differential element dt identi ed with an interval of resolution, regarded as a new variable The issue is not the description of what occurs in extreme cases, but the behavior of this function during successive zooms on the interval dt 1432 From continuity and non-differentiability to fractality It can be proved [BEN 00, NOT 93, NOT 96a] that the length L of a continuous and nowhere (or almost nowhere) differentiable curve is dependent explicitly on the resolution at which it is considered and, further, that L( ) remains strictly increasing and tends to in nity when 0 In other words, this curve is fractal (we will use the word fractal in this general sense throughout this chapter) Let us consider a curve (chosen as a function f (x) for the sake of simplicity) in the Euclidean plane, which is continuous but nowhere differentiable between two points A0 {x0 , f (x0 )} and A {x , f (x )} Since f is non-differentiable, there is a point A1 of coordinates {x1 , f (x1 )}, with x0 < x1 < x , such that A1 is not on the segment A0 A Thus, the total length becomes L1 = L(A0 A1 ) + L(A1 A ) > L0 = L(A1 A ) We can now iterate the argument and nd two coordinates x01 and x11 with x0 < x01 < x1 and x1 < x11 < x , such that L2 = L(A0 A01 ) + L(A01 A1 ) +L(A1 A11 ) + L(A11 A ) > L1 > L0 By iteration we nally construct successive
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approximations of the function f (x) studied, f0 , f1 , , fn , whose length L0 , L1 , , Ln increase monotonically when the resolution (x x0 ) 2 n tends to zero In other words, continuity and non-differentiability imply monotonous scale dependence of f in terms of resolution However, the function L( ) could be increasing but converge when 0 This is not the case for such a continuous and non-differentiable curve: indeed, the second stage of demonstration, which establishes the divergence of L( ), is a consequence of Lebesgue theorem (1903), which states that a curve of nite length is differentiable almost everywhere (see for example [TRI 93]) Consequently, a non-differentiable curve is necessarily in nite These two results, taken together, establish the above theorem on the scale divergence of non-differentiable continuous functions A direct demonstration, using non-standard analysis, was given in [NOT 93], p 82 This theorem can be easily generalized to curves, surfaces, volumes, and more generally to spaces of any dimension Regarding the reverse proposition, a question remains as to whether a continuous function whose length is scale-divergent between any two points such that x = xA xB is nite (ie, everywhere or nearly everywhere scale-divergent) and non-differentiable In order to prepare the answer, let us remark that the scale-dependent length, L( x), can be easily related to the average value of the scale-dependent slope v( x) 1 + v 2 ( x) Since we consider curves such that Indeed, we have L( x) = L( x) when x 0, this means that L( x) v( x) at large enough resolution, so that L( x) and v( x) share the same kind of divergence when x 0 Basing ourselves on this simple result, the answer to the question of the non-differentiability of scale-divergent curves is as follows (correcting and updating here previously published results [NOT 08]): 1) Homogenous divergence Let us rst consider the case when the slopes diverge in the same way for all points of the curve, which we call homogenous divergence In other words, we assume that, for any couple of points the absolute values v1 and v2 of their scale-dependent slopes verify: K1 and K2 nite, such that, x, K1 < v2 ( x)/v1 ( x) < K2 Then the mode of mean divergence is the same as the divergence of the slope on the various points, and it is also the mode of longitudinal divergence In this case the inverse theorem is true, namely, in the case of homogenous divergence, the length of a continuous curve f is such that: L in nite (ie, L = L( x) when x 0) f non-differentiable 2) Inhomogenous divergence In this case there may exist curves such that only a subset of zero measure of their points have divergent slopes, in such a way that the
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