If D is a dense subset of R and f is uniformly continuous, prove that f has a continuous extension to R.(adsbygoogle = window.adsbygoogle || []).push({});

I said:

if x0 is in D, then f(x0) is continuous.

Let x0 be in R\D. D dense in R--> there exists {x_n} in D s. t. {x_n}--> x0.

{x_n} is a Cauchy sequence converging to x. f is uniformly continuous on D--> {f(x_n)} is Cauchy and thus converges to y.

Define f(x0)=y

Then I'm stuck. I want to show that f(x0) is continuous, but I'm not sure how.

Another question: Prove that if D is closed (instead of dense) , then f also has a continuous extension

I'm not sure even where to begin with this question.

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# Homework Help: 2 questions on continuity/continuous extensions

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