現代数学の系譜工学物理雑談古典ガロア理論も読む48

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494: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/24(日) 10:30:54.63 ID:Q5UHveEY

>>493 つづき

---------------------------------------------------------------
[15] Gerald Arthur Heuer, "Functions continuous at the
irrationals and discontinuous at the rationals",
American Mathematical Monthly 72 #4 (April 1965), 370-373.
[MR 31 #3550; Zbl 131.29201]

Let f(x) = 0 if x is irrational, f(p/q) = |1/q| if
p and q are relatively prime integers, and f(0) = 1.

We say that a function g is Lipschitzian at x if there
exists a neighborhood U of x and a number M > 0 such
that |g(x) - g(y)| <= M*|x - y| for all y in U.

THEOREM 2: The function f^r is: (A) discontinuous at the
rationals for every r > 0; (B) continuous but
not Lipschitzian at the Liouville numbers, for
every r > 0; (C) differentiable at every irrational
algebraic number of degree <= r-1, if r > 3.

THEOREM 3: The function f^r is differentiable at every
algebraic irrational number if r > 2 (and, by
Theorem 1, at none if r <= 2).

THEOREM 4: The function f^2 is Lipschitzian but not
differentiable at the points of the set
{(1/2)*[m - sqrt(d)]: m is an integer
and there exists an integer n such that
d = m^2 - 4n is positive but not a perfect
square} . [This set is dense in the reals.]

THEOREM 5: If g is a function discontinuous at the
rationals and continuous at the irrationals,
then there is a dense uncountable subset
of the reals at each point of which g fails
to satisfy a Lipschitz condition.

つづく

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