[分析] 兩題跟uniformly continuous有關的題目

[king31815 (口八押係)]

這是我在zygmund實分析的第一章看到的題目
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1. If f is defined and uniformly continuous on E, show there is a function f
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   defined and continuous on E(closure) such that f=f on E.

2. If f is defined and uniformly continuous on a bounded set E, show that f

   is bounded on E.

這兩題我在rudin跟apostol上都有看到類似的習題，但是卻都是跟R^n空間有關，像是第一

題的codomain就是在R上，而第二題的domain是在R^n上，所以一個可以利用完備性，一個

可以利用Heine-Borel thm，就可以解決，所以也令我懷疑這兩題的命題是否有錯。

想麻煩高手為我解答了~拜託~

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◆ From: 124.8.9.175
推 zombiea     :use uniformly conti. sends chauchy seq to chauchy
→ king31815   :but I can not sure the limit existence...
推 lwei781     :1.f(x) = x, domain 取[0,1] 交 Q, Range 取Q
推 lwei781     :closure of E 當成是[0,1] in R with usualy metric
→ lwei781     :how to extend this?