Continuous Functions
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Let S be a subset of the metric space E with the property that each point of eS is a cluster point of S (one then calls S dense in E). Let E' be a complete metric space and f:S->E' a uniformly continuous function. Prove that f can be extended to a continuous function from E into E' in one and only one way, and that this extended function is also uniformly continuous.
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Solution Summary
Continuous Functions are explicated.
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Proof:
I extend the domain of from to . Since is dense in , then for any point , we can find a sequence , such that . Now I define . This is well-defined because is continuous and is complete.
First, I claim that this definition is ...
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