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I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds.

In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and if $f:U \cap V\rightarrow \mathbb R$ is a smooth function, then there exists two smooth functions $f_U, f_V$ defined on $U, V$ such that $f=f_V-f_U$. In order to do it, let $\{\rho_U, \rho_V\}$ be a partition of unity subordinate to the open cover $\{U, V\}$. Then we define $f_V(x)=\rho_U(x)f(x)$ if $x \in U \cap V$, or $0$ if $x \in V \setminus U \cap V $.

Similarly, we define $f_U$ and we are done. But first, we must show that $f_V$ and $f_U$ are smooth. It's clear that $f_V$ is smooth on $U \cap V$ and on the interior of $V \setminus U \cap V$, but I'm having trouble on understanding why $f_V$ is smooth on the frontier.

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You just need a hint: The support of $\rho_U$ (i.e. the closure of $\{ x\in M : \rho_U (x) \neq 0 \}$ in $M$) is a closed subset of $U$, so what can we say about the smoothness of $f_V=\rho_U \cdot f$ at any point in the complement of the support of $\rho_U$?

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