I was reading M,Golubitsky & V.Guillemin's book Stable Mappings and Their Singularities ,in theorem 2.7 (page 40)
Which construct the smooth structure on Jet bundle as follows:
Let $U$ be the domain for a chart $\phi$ on $X$ and $V$ be the domain for a chart $\psi$ on $Y$. Let $U^{\prime}=\phi(U)$ and $V^{\prime}=\psi(V)$. Then $\left(\phi^{-1}\right)^{*} \psi_{*}: J^{k}(U, V) \rightarrow$ $J^{k}\left(U^{\prime}, V^{\prime}\right)$ and $\tau_{u, v} \equiv T_{U^{\prime}, V^{\prime}} \cdot\left(\phi^{-1}\right)^{*} \psi_{*}: J^{k}(U, V) \rightarrow U^{\prime} \times V^{\prime} \times B_{n, m}^{k} .$ Give $J^{k}(X, Y)$ the manifold structure induced by declaring that $\tau_{u, v}$ is a chart. Which needs to prove transition is smooth then.
I found a post here that has exactly the same question,here.
My question is on the $J^{k}(U,V) \subset J^k(X,Y)$,if we take bump function multiply with $f$ which needs not to have compact support,which means it needs not coincide with $f$ on whole $U$?