注 ＊）f_w(0) = 1を書く意味は、0は無理数でもなく、p/qとも表せないということかな ＊＊）（>>285より抜粋） ** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25) (引用終り)
http：//mathforum.org/kb/message.jspa?messageID=5432910 Topic： Differentiability of the Ruler Function Dave L. Renfro Posted： Dec 13, 2006 Replies： 3 Last Post： Jan 10, 2007
これを読んでいて、疑問に思ったことが２点ある
１． ”[20] Marc Frantz, "Two functions whose powers make fractals", American Mathematical Monthly 105 #7 (Aug./Sept. 1998), 609-617. [MR 99g：28018； Zbl 952.28006] Following up on Darst/Taylor [18] above, Frantz investigates the Hausdorff dimension of the graphs of f^r.
THEOREM 1： If r > 2, then the Hausdorff dimension of the non-differentiability set for f^r is 2/r.”
一方、 ”[18] Richard Brian Darst and Gerald D. Taylor, "Differentiating Powers of an Old Friend", American Mathematical Monthly 103 #5 (May 1996), 415-416. [MR1400724； Zbl 861.26002]
Define f：R --> R by f(x) = 0 if x is irrational or zero, and f(p/q) = 1/q for p,q relatively prime with q > 0. They note that the set of points at which f is not continuous is the set of nonzero rational numbers.
THEOREM： If 1 < r <= 2, then f^r is differentiable only at zero. If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).”
だから、[18] からすると、If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).→Hausdorff dimension =1 で、"1 - 2/r（>>285）"ではないのでは？
