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Real analysis

  1. Nov 17, 2013 #1
    1. The problem statement, all variables and given/known data

    Let ##S\subset E## where ##E## is a metric space with the property that each point of ##S^c## is a cluster point of ##S.## Let ##E'## be a complete metric space and ##f: S\to E'## a uniformly continuous function. Prove that ##f## can be extended to a continuous function from ##E## into ##E'## and that this extended function is also uniformly continuous.

    2. The attempt at a solution

    This is a complicated problem for me and I want to know if my reasoning is correct?

    Let ##s\in S^c## then since ##s## is a cluster point of ##S## there exists a sequence ##s_n \in S## such that ##s_n \to s## thus ##s_n## is a Cauchy sequence. Since ##f## is a uniformly continuous function and ##E'## is complete then ##f(s_n)## is also a Cauchy sequence that converges in ##E'## to some ##x,## thus we have ##\lim f(s_n) = x.##

    Now we want to show that ##f## can be extended to ##E## and that this extension is uniformly continuous, in order to do that we must have that for any sequence in ##E## that converges to ##s## then the image sequence must converge to the same limit ##x.## Therefore we define our extension to be ##\lim g(y_n) = x## for ##y_n \in E.##

    Now to prove that this is indeed an extension we must show two things: that is uniformly continuous and that the image sequence converges to the same limit ##x## thus ##\lim g(y_n) = x.##

    Is my reasoning correct? Thank you for your time!
     
  2. jcsd
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