推 xx52002:感謝 Q<>Q 163.23.231.88 11/15 07:47
※ 引述《xx52002 (長門有希好萌(′▽‵))》之銘言：
: (1)Suppose the function f has the property that │f(x) - f(t)│≦│x-t│
: for each pair of points x,t in the interval (a,b).
: Prove that f is continuous on (a,b)
: (2)Given that f and g are continuous functions on [a,b], and the f(a)>g(a)
: and g(b)>f(b), show that there exists at least one number c 屬於 (a,b)
: such that f(c) = g(c)
: 拜託各位了 QQ
(1) 令 h→0 such that ｜f(x+h)－f(x)｜≦｜(x+h)－x｜＝｜h｜ (移項可得)
｜ f(x+h)－f(x) ｜
=> ｜lim ______________｜ ≦ １ => ｜f'(x)｜≦ 1 ∴ f'(x) 於(a,b)存在
｜h->0 h ｜
f(x)－f(t)
∴ lim f(x) ＝ lim｛〔 ____________ × (x－t)〕＋ f(t)｝＝f'(x)× 0 ＋f(t)
x->t x->t x－t
＝ f(t) 極限存在 for t in (a,b) ∴ f 連續 on (a,b)
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