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Math Help - Find a function that is defined and continuous on the closure of a set E.

  1. #1
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    Find a function that is defined and continuous on the closure of a set E.

    If f is defined and uniformly continuous on E, can somebody show me that there's a function g defined and continuous on the closure of E such that g = f on E?

    Since, for every limit point x of E, there is a sequence {xn} in E such that limxn = x, I define g as below.

    g(x)=f(x) if x in E; g(x)=limnf(xn) if x belongs to (the closure of E)\E.

    Would it be a right start? If so, then how do I prove that g is defined and continuous on (the closure of E)\E by using the fact that f is uniformly continuous?
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  2. #2
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    Re: Find a function that is defined and continuous on the closure of a set E.

    Quote Originally Posted by Rita View Post
    If f is defined and uniformly continuous on E, can somebody show me that there's a function g defined and continuous on the closure of E such that g = f on E?
    You should have a lemma that says that if (x_n) is a Cauchy sequence in E then f(x_n) is a Cauchy sequence. And that last sequence is converges to some b\in\overline{E}.

    More over if both (x_n)~\&~(y_n) are Cauchy sequences in E and (x_n)\to b~\&~(y_n)\to b then f(x_n)\to f(b)~\&~f(y_n)\to f(b). Now that gives you a natural way to define the extension of f .
    Thanks from topsquark and Rita
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