$\mathbf {The \ Problem \ is}:$ Given , $U$ is an open subset of $\mathbb R^{m+n}$ and $f : U \to \mathbb R^n$ be a $C^1$ map ,then show that the map $F : (p,v) (\in U ×\mathbb R^{m+n}) \mapsto Df_p (v) (\in \mathbb R^n)$ is continuous .
$\mathbf {My \ approach}:$ Actually, taking a nbhd $V$ of some point $Df_p (v)$ for some $p,v$ in $U$ and $\mathbb R^{m+n})$ resp. ,then it's getting difficult for me to think of nbhd $N$ around $(p,v)$ such that $F(N)$ is contained in $V.$
I am trying to use inverse function theorem, but there was a hypothesis that at some point $a$,$Df_a$ needs to be an isomorphism .
If $Df_p$ is onto for some $p \in U$, then $f$ locally ''looks like a'' a projection map.
Is this question correct? If it is, then a small hint is appreciated .
Thanks in advance .
Hint
For $r,s \in \mathbb N$, let $\mathcal L(\mathbb R^{r}, \mathbb R^s)$ be the linear space of linear maps from $\mathbb R^r$ to $\mathbb R^s$.
The map
$$\begin{array}{l|rcl} G : & \mathcal L(\mathbb R^{m+n}, \mathbb R^n) \times \mathbb R^{m+n} & \longrightarrow & \mathbb R^n \\ & (U,v) & \longmapsto & U(v) \end{array}$$ is a bilinear map defined on a finite dimensional linear space. It is therefore continuous (and even smooth).
As $f$ is supposed to be $\mathcal C^1$ and
$$F(x,v) = G(f^\prime(x),v)$$ we get that $F$ is continuous as a composition of continuous maps.