Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is actually $\mathcal{C}^{\infty}$ on $M-K$ for some closed set $K \subset M$. Let $\varphi$ be a continuous extention of $h$ in $X^{(1)}$, i.e., $p^{1}_{0} \circ \varphi=h$. Let $\mathcal{R}$ be a neighborhood of $\varphi$ in $X^{(1)}$.
Then I consider the bundle $\pi: \iota^{*}X^{(1)} \rightarrow h(M)$, which is the pull-back bundle of $\pi: X \rightarrow M$ along the map $\iota: h(M) \rightarrow X$. Note $\varphi$ can be identified as a section in $\pi: \iota^{*}X^{(1)} \rightarrow h(M)$ by the map(denoted by $\overline{\varphi}$) $h(x)\mapsto\varphi(x)$ for $x\in M$. So $\overline{\varphi}(X) \subset \mathcal{R}\cap\iota^{*}X^{(1)}$.
My question is that does there is some smooth approximation result that can be used to pertube $\overline{\varphi}$ in the neighborhood $\mathcal{R}\cap\iota^{*}X^{(1)}$ so that the resluting section is smooth over $h(M) - h(K)$ and remains the same over $K$?
Straightly, I want to apply standard approximation theorem over the $\mathcal{C}^{\infty}$manifold $h(M)-h(K)$. But the question now leads to how to patch together with the one over $h(K)$(which is closed in $h(M)$) and get the coresponding homotopy rel $h(K)$?
Thank you.