さて、定理1.7 (422 に書いた定理)のそもそもの目的は、変形トマエ函数（Ruler Function）関連で、 「系1.8 有理数の点で不連続、 無理数の点で微分可能となるf : R → R は存在しない」を導くことであった
変形トマエ函数（Ruler Function）関連については、過去スレで取り上げているが、いま一度整理すると （長いが、あとのために抜粋する） 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) （注：下記で、f^rなどとして、ｒの指数による類別をしている） 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.
It is well-known that f is continuous at each irrational point and discontinuous at each rational point.
** For each r > 2, f^r is differentiable on a set that has c many points in every interval.
The results above can be further refined.
** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]
** 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]
