0371現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE
2018/10/18(木) 11:49:57.25ID:iaTtGtuuつづき
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]
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.
つづく