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。）
[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)
ああ、いま改めて読むと Bulletin of the Calcutta Mathematical Society 49 (1957) Senguptaより ”・・・ 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).”
なんてありますね。”at least one of the four Dini derivates of f is infinite”が、貴方の定理に近いかな？ ”Then E is co-meager in R (i.e. the complement of a first category set).”か・・ これか、これに近い文献を読まないことには、訳わからんな
えーと、Meagre setか・・ ”E is co-meager in R”が、イメージできんな・・（＾＾
前提a)(連続不連続が稠密)を、b)(連続とディニ微分発散が稠密な組み合わせ)に、緩和しても・・ a) f is continuous and discontinuous are each dense in R. ↓ b) f is continuous and the E *) are each dense in R. ( *)the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.)
