I'm trying to solve the following problem from the book "Semi-Riemannian Geometry" by Barrett O'Neill (first edition):
A vector field $V$ on M is smooth if for sufficiently many coordinate system $\xi$ to cover M the functions $Vx^i$ are smooth
My attempt:
Let A be a complete atlas on M, i.e., any $\xi \in A$ intersects all $\eta \in A$. As $\xi$ is a homeomorphism from $U \subset M \rightarrow R$, we have that $\xi$ and $\xi^{-1}$ are continuous. Then, if $\xi=(x^1,...,x^i)$, $x^i$ must be continuous.
Let $\partial_i$ be a vector field on U and $p \in U$. So $\partial_i x^i=\frac{\partial_i x^i}{\partial_i x_i}$ which is smooth.
Note that $V$ can be locally written as $V=a_1X_i+...+a_nX_n$, where $a_i’s$ are (not necessarily smooth) maps on the manifold and $X_i$ is the local vector field corresponding to the $i$th coordinate of the local chart. Since $V(x_i)$ is smooth for all $i$ by hypothesis, each $a_i$ which is equal to $V(x_i)$ is smooth. Hence $V$ is smooth.