[分析] 一道一致連續題目
題目:Let f(x) be continuous in a finite interval (a,b). Prove that f(x) can be continuously extended to [a,b] if and only f(x) is uniformly continuous in (a,b).
我的解答是:
Only if part(=>)
If f(x) can be continuously extended to [a,b], f(x) is continuous on [a,b]. Therefore, f(x) is uniformly continuous on [a,b]. So, f(x) is also uniformly continuous on (a,b).
If part(<=)
If f(x) is uniformly continuous on (a,b), we let {x_n} be sequence in (a,b) that converges to a. Since {x_n}is a Cauchy Sequence, f(x_n)is a Cauchy Sequence too.
Therefore, lim(x->a)f(x)=p
Similarly, lim(x->b)f(x)=q
Therefore, f(x) can be continuously extended to [a,b].
我想問的是我的答法對是不對, 有沒有改進的餘地如寫得簡潔一點?
謝謝各位的指導.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 112.119.153.89
推
11/10 00:38, , 1F
11/10 00:38, 1F