I am trying to prove the existence of a local transversal section, i.e.
Let $U\subset\mathbb R^n$ be an open, $X:U\to \mathbb R^n$ a vector field of class $C^k$ and $p\in U$ a regular point of $X$, then there exists a local trasversal section $f$ of class $C^\infty$ to the field $X$ at point $p$ such that $f(0)=p$.
My attempt was the following: Since $X(p)\ne 0$, then there are vectors $v_1,\dots,v_{n-1}\in\mathbb R^{n}$ such that $\beta_p=\{v_1,\dots,v_{n-1},X(p)\}$ is a base of $\mathbb R^n$. I managed to prove that there exists $\delta>0$ such that $\beta_q$ is a base of $\mathbb R^n$ for all $q\in B(p,\delta)$. Taking $M=\max\{\|v_j\|\}$ and $\epsilon<\delta/M$, we defined $f:A=\{x\in\mathbb R^{n-1}:\|x\|<\epsilon\}\to\mathbb R^n$ by $f(x)=p+\displaystyle\sum_{j=1}^{n-1}x_jv_j$. It is easy to see that $f$ is $C^\infty$ and $Df(x)$ is injective for all $x$. It's also easy to see that $f$ is a bijection, but I don't know how to prove that it has a continuous inverse. The fourth condition comes out easily once we notice $Df(x)(e_j)=v_j$. Some help please.
Let $U\subset\mathbb R^n$ be an open, $X:U\to \mathbb R^n$ a vector field of class $C^k$. $\color{red}{\text{A local transversal section of class }C^k\text{ to the field }X\text{ at }p\in U}$, is a function $f:A\to U$ of class $C^k$ where $A$ is a connected open of $\mathbb R^{n-1}$, which satisfies: 1) $f$ is a immersion. 2) $f:A\to \Sigma=f(A)$ is a homeomorphism. 3) $p\in\Sigma$. 4) $Df(x)(\mathbb R^{n-1})\oplus Span\{X(f(x))\}=\mathbb R^n$ for all $x\in A$.