18/02/25 17:20:54.82 r2nQOARz.net
>>359 補足
>Define x(n) = 1/√(2πn + π/2) and y(n) = 1/√(2πn)
これちょっと怪しい（＾＾
下記が正解（√が不要）だろう
URLﾘﾝｸ(www.eco.uninsubria.it)
Quaderni di ricerca (2000 / 2016)
URLﾘﾝｸ(eco.uninsubria.it)
I. Ginchev Weakened subdifferentials and Frechet differentiability of real functions 2008/3
Universita degli Studi dell'Insubria Via Monte Generoso, 71, 21100 Varese, Italy
(抜粋)
P5
Example 2. The function f : R → R,
f(x) =x^3/2 sin(1/x) , x ≠ 0 ,
=0 , x = 0 ,
is not Lipschitz near x = 0, but it possesses a Lipschitz weakened derivative f^w(0, v) = 0.
To show that the function f in this example is not Lipschitz near x = 0 put
xn = 1/{(2n ? 3/2)}π, yn =1/(2nπ).
Then xn → 0, yn → 0 and
{f(xn) ? f(yn)}/(xn ? yn) = 4n/{(3π)^(1/2)(2n ? 3/2)^(1/2)}→ ∞ as n → ∞.
We conclude this section with an example showing that the finiteness of L_*f^w(x) does not
imply the finiteness of of L*f^w(x) and consequently the Lipschitz property of f^w(x, ・).
(引用終り)