>>370
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