Suppose we have a (say compactly supported) $C^0$-vector field $X:\mathbb R^n\to\mathbb R^n$ such that for every $x\in\mathbb R^n$ there is a unique $C^1$-curve $\gamma:\mathbb R\to\mathbb R^n$ solving $\dot\gamma_x(t)=X(\gamma_x(t))$ with $\gamma_x(0)=x$.

Then the ode flow $\mathcal F_X$ is pointwisely defined in the way that $\mathcal F_X(t,x)=\gamma_x(t)$.

My **question** is: Must $\mathcal F_X$ be a continuous map with respect to $x$?

Certainly there is no problem if $X$ is Lipschitz or just satisfies the Osgood condition. Since we have the regularity estimate of ODE flows with respect to their modulus of continuity.

When $X$ is H"older and somehow its ODE is uniquely solvable at every point, could their be continuous dependence? And if not how does the blow up occur.