Suppose that I have a well-defined smooth vector field $Y$ of the form $\sum_i f_iX_i$ on a submanifold $S$ of $M$ where each $f_i$ is a smooth function on $S$ and each $X_i$ is a smooth vector field on $S$.
Suppose also that I know $f:= \sum_i f_i$ cannot be continuously extended to all of $M$. Does this imply that $Y$ can also not be extended to all of $M$?
I don't know whether extending $Y$ would need to extend $f$ in an impossible way. Any help is appreciated.