Real Analysis : Continous Extension Theorem
A: Show that a uniformly continous function preserves Cauchy sequences; that is, if f:A->R is uniformly continous and (x_n) subset or equal of A is a Cauchy sequence then show f(x_n) is a Cauchy sequence.
B: Let g be a continous function on the open interval (a,b). prove that g is uniformly continous on (a,b) if and only if it is possible to define values g(a) and g(b) at the endpoints so that the extended function g is continous on [a,b]. (In the forward direction, first produce candidates for g(a) and g(b) and the show the extended g is continous).
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Solution Summary
The Continous Extension Theorem and Cauchy sequances are investigated. The solution is detailed and well presented.
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