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Let $E$ be a closed set in $\mathbb{R}$.

Then $E^c$ is open, hence it is a union of at most countable collection of disjoint segments, $\{I_i\}$.

Say, $I_i=(a_i,b_i)$.

Now, suppose $x\in E$ and $x$ is not an interior point of $E$.

How do i prove that $x$ is an endpoint for some $I_i$?

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    You can't. See [this](http://math.stackexchange.com/q/210735/8348) related question.2012-10-23

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The simplest example I can come up with to show that this is not the case is the following: Let $E = \left\{ \frac{-1}{n} : n \in \mathbb{N} \right\} \cup \{ 0 \} \cup \left\{ \frac{1}{n} : n \in \mathbb{N} \right\}.$ Clearly $E$ is closed, and its complement is $( - \infty , -1 ) \cup \bigcup_{n=1}^\infty \left( \frac{-1}{n} , \frac{-1}{n+1} \right) \cup \bigcup_{n=1}^\infty \left( \frac{1}{n+1} , \frac{1}{n} \right) \cup ( 1 , + \infty ).$ However $0$ is not the endpoint of any of these intervals.

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    @Katlus: I haven't gone through the details, but the only outstanding case is if the point $p$ is an accumulation point of endpoints of the open intervals, and hence also of $E$ itself. You should then be able to use the fact that the original function $f$ is continuous on $E$ together with the fact that $g$ is linear on the open intervals $(a_i , b_i)$ to show that $g$ is continuous at $p$.2012-10-23