※ 引述《kinki1843 (onizuka)》之銘言：
: ※ 引述《sdsa (加油)》之銘言：
: : (a)Is there a continuous function f from [0,1] onto (0,1)?Why?
: : (b)Is there a one-to-one function g from [0,1] onto (0,1)?Why?
: : 謝謝指教了
: : 或許這題對有些人來說很簡單
: : 但是我的高微程度真的很差
: : 可以耐心幫我講解嗎
: : 我觀念不是很清楚
: : 都沒有人可以問 嗚嗚
: (a)no!(0,1)不為compactㄉ,連續函數保有compact
: (b)是吧,應該是自己第ㄉ含數為onto就可以了
: over!
(a) Theorem 9.29 (William Wade third edition)
if H is compact in R^n and f: from H to R^m is conti
then F(H) is compact in R^n
since [0,1] is compact , if f is conti from [0,1] to (0,1)
,then (0,1) must be compact. There is a contradiction.
(b) f(x)= 1/2 when x=0
1/n+2 when 1/x belongs to N
( eg, f(1/1)= 1/3 f(1/2)=1/4 )
x otherwise
then f: [0,1] to (0,1) is a 1-1 function
據說是數導題目?
--
http://www.wretch.cc/album/savory
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 210.85.38.109