Let $U\subset \mathbb{R}^n$ be an open subset and let $X:U\to\mathbb{R}^n$ be a vector field over $U$. Also let $I\subset\mathbb{R}$ be an interval and $\gamma:I\to U$ be an integral curve satisfying the initial condition $\gamma(t_0)=x_0$ where $t_0\in I , x_0\in U$ are given. (existence of such integral curve is guaranteed by Peano existence theorem provided that $X$ is $C^0$)
Then the following definition is given:
$X$ is said to have uniformity of integral curves with respect to initial conditions if there exists an open neighborhood $x_0\in V\subseteq U$ such that for all $x\in V , J\subsetneq I\in\mathbb{R}$ (arbitrary interval), there exists an integral curve $\gamma_x:J\to U$ satisfying the initial condition $\gamma_x(t_0)=x$.
My question is that under what circumstances does this property hold and why?
In my textbook it is noted that this property holds for differentiable vector fields (without a proof) but due it's probably an overpowered condition. I was looking forward to some kind of Lipschitz continuity condition but failed to get to a proof.
My attempts were mostly trying to use some kind of fixed point theorem. Let $f:I\to\mathbb{R}^n$ be an arbitrary continuous (hence Riemann integrable) function. Then one could deduce:
$$X(f(t))=f'(t), \forall t\in J \iff\int_{t_0}^t X(f(s)).ds=\int_{t_0}^tf'(s).ds=f(t)-f(t_0)$$
Therefore if we define $F_x(f)=\displaystyle\int_{t_0}^t X(f(s)).ds+x\in C^1$, we're actually trying to prove that $F_x$ has a fixed point in the space of differentiable functions $f:J\to\mathbb{R}^n$.
I first tried to use Schauder fixed point theorem, but failed trying to prove that the space of differentiable functions $J\to\mathbb{R}^n$ is a compact subspace of the Banach space of continuous functions $J\to\mathbb{R}^n$, equipped with the supremum norm.
Next, I tried using Banach fixed point theorem. But $F_x$ being a contraction is equivalent to $L.l(J)<1$ where $l(J)$ is length of the interval $J$ and $L$ is the Lipschitz constant of $X$. (assuming it is Lipschitz continuous) But I don't think that we can assume such inequality to be true under minor modifications to the problem.
I've also tried minor tricks such as trying to extend the local integral curves satisfying $\gamma_x(t_0)=x_0$ but I cannot get to a valid proof.
Any help/suggestion/source to a proof would be appreciated!