High level people は自分達で勝手に立てたスレ28へどうぞ!sage進行推奨(^^; また、スレ43は、私が立てたスレではないので、私は行きません。そこでは、私はスレ主では無くなりますからね。このスレに不満な人は、そちらへ。 http://rio2016.2ch.net/test/read.cgi/math/1506152332/ 旧スレが512KBオーバー(又は間近)で、新スレ立てる (スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。)
まだ、疑問に思っているのは 下記のDifferentiability of the Ruler Functionの記述と貴方の定理との整合性だ
http://mathforum.org/kb/message.jspa?messageID=5432910 (>>35より) Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) 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.
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define 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.
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
[13] Gerald Arthur Heuer, "Functions continuous at irrationals and discontinuous at rationals", abstract of talk given 2 November 1963 at the annual fall meeting of the Minnesota Section of the MAA, American Mathematical Monthly 71 #3 (March 1964), 349.
The complete text of the abstract follows, with minor editing changes to accommodate ASCII format.
Earlier results of Porter, Fort, and others suggest additional questions about the functions in the title. Differentiability and Lipschitz conditions are considered. Special attention ispaid to the ruler function (f) and its powers. Sample results: THEOREM: If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere differentiable, but is Lipschitzian on a dense subset of the reals. THEOREM: If r > 0, f^r is continuous but not Lipschitzian at every Liouville number; if r > 2, f^r is differentiable at every algebraic irrational. THEOREM: If g is continuous at the irrationals and not continuous at the rationals, then there exists a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO: The last theorem follows from the following stronger and more general result. Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below. (引用終り)