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Let $C$ be closed set in $\Bbb R$ and let $f:C\to \Bbb R$ be continuous.

Show that there is a continuous function $g:\Bbb R\to \Bbb R$ with $g(x)=f(x)$ for every $x\in C$.

I am unable to view how to construct $g$ from $f$.

Please give some hints.

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  • Think about a function which extends what happens on the boundary of $C$. – gt6989b Dec 13 '16 at 16:25
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    This is a direct consequence of Tietze's extension theorem https://en.wikipedia.org/wiki/Tietze_extension_theorem – Dominik Dec 13 '16 at 16:26
  • Since $C$ is closed then the complement is open in $\mathbb{R}$ so it is a countable union of intervals. I think (perhaps incorrectly) you use this to extend the function linearly inside each interval. – user71352 Dec 13 '16 at 16:30

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Have a look at Tietze extension theorem.