※ 引述《kgbtdaguo (daguo)》之銘言：
: let f(x)=x^2 *(e)^(x^2) for all x屬於R
: f^-1 exist and is differentiable on (0,無限大)
首先，易知 f'(x) ＞ 0 on (0,∞), 此導致 f 為嚴格遞增(且連續)函數 on (0,∞).
因此，f 之反函數 f^(-1), (稱 g) 存在 on (0,∞).
再來，欲證 g 為可微函數 on (0,∞): 這個我就不寫了…給你資料:
http://frankmath.cc/plover/Apostol.pdf p. 188.
NOTE. 請與此題作一比較…
反函數定理的一個假設條件 "the continuity of f' at a point a" 不能省。
Show that the continuity of f' at a point a is needed in the inverse function
theorem, even in the case n＝1：If
f(t) ＝ t ＋ 2t^2 sin (1/t) if t ≠ 0,
＝ 0 if t ＝ 0.
then f'(0)＝1, f' is bounded in (－1,1), but f is not 1-1 in any neighborhood
of 0.
[Ref. Walter Rudin, Principles of Mathematical Analysis, 3rd ed. McGraw-Hill,
1976. Exercise 16, Chap. 9, p. 241.]
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※ 編輯: math1209 來自: 220.133.4.14 (01/07 00:24)