>>34

つづき

(2) Suppose we alter the definition of g so that 2^q
is replaced by w(q), where w:Z+ --> Z+ is some increasing
function. Then the following are left to the reader.
(See [Nymann's paper] for (a) and other related results.)
(a) If w(q) = q^2, then g is nowhere differentiable.
(Use (2).)
(b) If w(q) = q^3, then g is differentiable on a dense,
uncountable set of irrationals, but nowhere twice
differentiable.
(c) No matter how rapidly w increases, the set A_0
of points of nondifferentiability is residual.

As a consequence of (c), no function vanishing at the
irrationals and discontinuous at the rationals can be
differentiable at the irrationals. In fact, a little
more argument shows that no function can be discontinuous
at every rational but differentiable at every irrational.
(This last has been known, by another method of proof,
for some time, e.g. [Boas' "Primer of Real Functions"],
[Fort's paper].) The following theorem implies (c) and
the above statements, and provides a nice application
of the Diophantine approximation point of view. (A slightly
weaker version appears in [Heuer's 1966 paper] and is
considered from a more general viewpoint in [Beesley,
Morse, and Pfaff's 1972 paper].)

つづく