When is the solution to an ODE $\dot x = f(x)$ uniformly continuous? 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.?) 
 A: I will only consider an initial value problem on $[0,\infty)$ for simplicity. Also let's assume that $f$ is continuous and defined on all $\mathbb{R}^n$. 
The solution is obviously uniformly continuous on any closed bounded interval on which it exists. 
Let $I$ be any interval containing $0$ on which the solution is uniformly continuous, then it may be extended to the closure of the interval and then the solution can be extended to a strictly larger interval. 
If the right hand side $f(x(t))$ is uniformly bounded for all $t$ in the maximal interval of existence, then the solution exists for all $t > 0$ and is uniformly continuous there.
But if the solution exists for all $t>0$, it may or may not be uniformly continuous on this set (example $x' = x$). 
In short, uniform continuity of the solution on its maximal interval of existence implies global existence but not the other way around. And  uniform continuity of the solution on its maximal interval of existence is essentially equivalent to uniform boundedness of the solution. 
