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I am wondering the following:

Let $M$ be a smooth manifold and let $U \subset M$ be an open subset. Is every map $f \in C(U)$, where $C(U)$ is the $\mathbb{R}$-vector space of smooth functions $g:U \rightarrow \mathbb R$, a restriction of a map $g \in C(M)$? Is this trivial to see or do I just want to prove something wrong?

Thanks!

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  • $\begingroup$ use partitions of unity i guess $\endgroup$ – M. Van Jan 27 '17 at 0:51
  • $\begingroup$ Partitions of unity allows me to create a function in $C(M)$ which is equal to $f$ if the subset $U$ is closed, not open. $\endgroup$ – Ale Jan 27 '17 at 0:52
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    $\begingroup$ This is false even for M equal to the real line and U the interval (0,1), as the function 1/X shows. $\endgroup$ – Mariano Suárez-Álvarez Jan 27 '17 at 0:58
  • $\begingroup$ The ask-a-question page should force users to make a list of at least 5 examples they have considered... $\endgroup$ – Mariano Suárez-Álvarez Jan 27 '17 at 0:59
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    $\begingroup$ (this is in fact false for all M and all U, provided U is a non-empty proper open subset of M) $\endgroup$ – Mariano Suárez-Álvarez Jan 27 '17 at 1:03

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