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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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    use partitions of unity i guess – 2017-01-27
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    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. – 2017-01-27
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    This is false even for M equal to the real line and U the interval (0,1), as the function 1/X shows. – 2017-01-27
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    The ask-a-question page should force users to make a list of at least 5 examples they have considered... – 2017-01-27
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    (this is in fact false for all M and all U, provided U is a non-empty proper open subset of M) – 2017-01-27
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    @MarianoSuárez-Álvarez: Actually, there's one situation in which it's true -- if $U$ is a union of connected components of $M$. – 2017-01-27
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    @jack, that's true — but my Ms are most connected ;-) – 2017-01-27

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