17/12/11 23:43:22.87 H5YTMI7H.net
>>566 つづき
NOTE: Sengupta/Lahiri had essentially obtained this result
in 1957 (the points of discontinuity have to form an
F_sigma set, however). See my remark in [13] above.
This result is also proved in Gerald Arthur Heuer,
"A property of functions discontinuous on a dense set",
American Mathematical Monthly 73 #4 (April 1966),
378-379 [MR 34 #2791]. Heuer proves that for each
0 < s <= 1 and for each f:R --> R such that
{x: f is continuous at x} is dense in R and
{x: f is not continuous at x} is dense in R,
the set of points where f does not satisfy a
pointwise Holder condition of order s is the
complement of a first category set (i.e. a co-meager
set). By choosing s < 1, we obtain a stronger version
of Sengupta/Lahiri's result. By intersecting the
co-meager sets for s = 1/2, 1/3, 1/4, ..., we get
a co-meager set G such that, for each x in G, f does
not satisfy a pointwise Holder condition at x for
any positive Holder exponent. (Heuer does not
explicitly state this last result.) A metric space
version of Heuer's result for an arbitrary given
pointwise modulus of continuity condition is essentially
given in: Edward Maurice Beesley, Anthony Perry Morse,
and Donald Chesley Pfaff, "Lipschitzian points",
American Mathematical Monthly 79 #6 (June/July 1972),
603-608 [MR 46 #304; Zbl 239.26004]. See also the last
theorem in Norton [17] below.
(引用終り)
以上