Consider a differentiable manifold $M$ and the smooth push-forward $d\phi : TM \to \mathbb{R}^n \times \mathbb{R}^n$ derived from the transition mapping $\phi$. Can we say anything about the continuity of the mapping $x \to d \phi(x)$ viewed as map from $M \to L(T_xM,\mathbb{R}^n)$ where $L(T_xM,\mathbb{R}^n)$ is the set of linear mappings between the tangent space $T_xM$ and $\mathbb{R}^n$
The reason i want to prove this is because i want to bound the operator norm over a compact set of $d\phi(x)$, so that if $K$ is compact set of $M$, then $$\sup_{x\in K} \|d\phi(x)\|< \infty$$
Thanks in advance.
I interpret the transition mapping to be a chart : $\phi : U \to \Bbb R^n$. Since $U$ is diffeomorphic to an open set in $\Bbb R^n$ we have canonically $TU \cong U \times \Bbb R^n$, the map your are looking for is $(x,v) \mapsto (\phi(x), d\phi(v))$ which is definitely smooth. (Your question seems not to use the Riemannian structure, since the same argument works as long as $M$ is a differentiable manifold.)