In many applications, it is highly desirable to know when the solution $x(t)$ to an ODE $\dot x = f(x)$, is uniformly continuous.
However, there seems to be very few words written on conditions on $f$ when this is true.
In general, the uniform continuity of $x(t)$ is given by the Caratheodory existence theorem (https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem). But this is a very general theorem that works for discontinuous functions $f$, and the conditions are a bit difficult to check. In any case, the existence theorem provides conditions on $f$ such that $x(t)$ is absolutely continuous, hence uniformly continuous. Is there a way to conclude uniform continuity without resorting to absolute continuity?
Can we conclude uniform continuity of the solution directly from Cauchy-Lipschitz or Peano's existence theorem?
What are some "nice" conditions on $f$ that allows us to immediately conclude $x(t)$ is uniformly continuous? (i.e. suppose $f$ is Lipschitz, then is $x(t)$ u.c.?)