>>484 誤爆スマン（＾＾

>>284-285 補足

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？

つづく