Vtyjkyuk

Let $mathcalF$ be a codimension-1 continuous foliation of $mathbbR^2$ with $C^1$ leaves. That is, a partition of $mathbbR^2$ into $C^1$ curves which can locally be mapped continuously to parallel lines. Define a distance along a leaf $d_mathcalF: mathbbR^2 times mathbbR^2 to mathbbR$ by: If $x$ and $y$ lie on the same leaf of the foliation, then $d_mathcalF(x,y)$ is the length of the leaf segment between them, and if they lie on different leaves, set $d_mathcalF(x,y)=infty$.

I want to show:

If the leaves of $mathcalF$ are tangent to a uniformly continuous
vector field $X$ on $mathbbR^2$, then the map
$d_mathcalF(x,y)$ is uniformly continuous with respect to the
standard metric on $mathbbR^2timesmathbbR^2$.

This arises in articles and books I have read about dynamical systems, often in more complicated contexts, but I'm using this setting at a starting point. However, I do not have a background in foliations, and I have just been trying to use basic differential geometry to show this.

It suffices to show that given $varepsilon>0$, there is $delta>0$ so that $|x-y| <delta implies |d_mathcalF(x,y)|<varepsilon$. However I struggle to proceed: I first try and show the result for $d_mathcalF$ restricted to a leaf, where I use a parametrisation of this leaf and use the definition for distance of a curve. However, I cannot manage to relate this to the property that the leaf is tangent to the vector fied $X$. If I assume the result holds when restricted to a leaf, I expect to use a foliation chart to extend the result to $mathbbR^2$, however I am also unsure of how this argument would follow.