18/10/25 23:53:20.43 AJCjq/E6.net
>>568
つづき
それは、下記だがね
（>>370より）
URLﾘﾝｸ(mathforum.org)
Topic: Differentiability of the Ruler Function
The Math Forum
Dave L. Renfro Registered: 12/3/04
(抜粋)
THEOREM: Let g be continuous and discontinuous on sets
of points that are each dense in the reals.
Then g fails to have a derivative on a
co-meager (residual) set of points. In fact,
g fails to satisfy a pointwise Lipschitz
condition, a pointwise Holder condition,
or even any specified pointwise modulus of
continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
[13] Gerald Arthur Heuer
REMARK BY RENFRO: The last theorem follows from the following
stronger and more general result. Let f:R --> R be such that
the sets of points at which f is continuous and discontinuous
are each dense in R. Let E be the set of points at which f
is continuous and where at least one of the four Dini derivates
of f is infinite. Then E is co-meager in R (i.e. the complement
of a first category set). This was proved in H. M. Sengupta
and B. K. Lahiri, "A note on derivatives of a function",
Bulletin of the Calcutta Mathematical Society 49 (1957),
189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in
item [15] below.
(引用終り)
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