推 waytin :假設g(x)=f(x) on (a,b) g(a)與g(b)是f(x)的極限值 12/13 11:51
→ waytin :這樣g(x)在[a,b]就連續，因此均勻連續。 12/13 11:53
→ waytin :不知道這樣可不可以？ 12/13 11:53
不好意思 我不太明白你這段證明 能否寫詳細點?@@"
※ 編輯: Helilo 來自: 99.135.74.143 (12/13 12:22)
Suppose f(x) is continuous on (a,b). Prove that f(x) is uniformly continuous
on (a,b) if and only if lim f(x) and lim f(x) exist and both are finite.
x->a+ x->b-
目前只想到 "==>" 不知道對不對
Proof:
Since f(x) is uniformly continuous, for any ε>0, exist δ>0
such that |x-y|<δ implies |f(x)-f(y)|<ε for any x,y in (a,b)
Let t_n be any sequence which converges to a from the right-hand side
=> there exists N such that |t_m - t_n|<δ for m>=N, n>=N
=> |f(t_m) - f(t_n)|<ε for m>=N, n>=N
=> f(t_n) converges to some p
=> lim f(x) = p
x->a+
Similarly for lim f(x)
x->b-
"<=="想不出來
還麻煩各位幫忙orz
另外"==>"有錯也煩請指正<(_ _)>
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