# Is every smooth map on an open the restriction of a smooth map

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!

• use partitions of unity i guess – M. Van Jan 27 '17 at 0:51
• 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. – Ale Jan 27 '17 at 0:52
• This is false even for M equal to the real line and U the interval (0,1), as the function 1/X shows. – Mariano Suárez-Álvarez Jan 27 '17 at 0:58
• The ask-a-question page should force users to make a list of at least 5 examples they have considered... – Mariano Suárez-Álvarez Jan 27 '17 at 0:59
• (this is in fact false for all M and all U, provided U is a non-empty proper open subset of M) – Mariano Suárez-Álvarez Jan 27 '17 at 1:03