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（簡単に要約すると）
The ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/q if x = p/q where p and q are relatively prime integers with q > 0.
で
１）r=0 は、ディリクレの関数で、いたるところで不連続
２）r=1 は、トマエ数で、有理数Qで不連続、無理数で連続
３）0 < r <= 2, f^r is nowhere differentiable
４）For each r > 2, f^r is differentiable on a set that has c many points in every interval.
５）For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition.
６）For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals
７）For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r
８）Using ruler-like functions that "damp-out" quicker
than any power of f
　f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.
　（increasing function that eventually majorizes every power function）
　f_w is differentiable on a set whose complement has Hausdorff dimension zero.
９）Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
（要約終わり）
＜前振り終わり＞

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