※ 引述《tandem (天燈)》之銘言：
: 證 [0,1] -> [0,1] 的遞增函數 f 必有一不動點 x in [0,1] , f(x) = x
: ---
: 沒有連續性, 不能用中間值定理
: 要怎麼做呢 @@
take any x in [0,1], the sequence x,f(x),f^2(x),... is always monotone
thus lim f^i(x) exists.
Let aij = f^i(f^j(0)), then lim lim aij, lim lim aij, lim aij all exist,
i j j i i,j
so lim lim f^i(f^j(0)) = lim lim f^i(f^j(0))
i j j i
lim f^i(a) = a, where a:=lim f^j(0).
i j
since a,f(a),f^2(a),... is monotone => a=f(a)=f^2(a)=...
ie. a is a fixed pt.
--
※ 編輯: Sfly 來自: 131.215.6.92 (12/03 10:53)