As to the border case. An example that you might like to consider is given by the blanc-mange curve, $f_{lambda}:mathbb{R}tomathbb{R} $, that for any value of $0leqlambdaleq 1,$ is defined as the unique bounded solution of the fixed point functional equation
$f(x)=mathrm{dist }(x,mathbb{Z})+lambda f(2x)$
(by the contraction principle there is a unique such function; it is continuous and 1-periodic, with an immediate series expansion coming from the iteration).
Consider $f_{lambda}$ on the unit interval. If you take $lambda=1/4$ you find a parabola; with $lambda < 1/2$ it's Lipschitz (hence the graph has a finite length) with constant $(1-2lambda)^{-1}$; if $lambda > 1/2$ it's Hoelder continuous with an exponent depending from $lambda$. The parameter 1/2 is critical: you find a curve that is not Lipschitz, but it's Hoelder of all exponents $alpha>0$, precisely, it has a modulus of continuity ct|log(t)|, and looking at it a bit more closely, it is not of bounded variation on any nontrivial interval (so the graph is not rectifiable even locally), nor is BV for any $lambda geq 1/2.$
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