Given the ODE:
\begin{align} \dot{x}(t) = f(x(t)) \end{align}
It is known that the solution $x(t)$ is continuously differentiable on a compact interval $t \in [0, t_{1}]$ when $f(x(t))$ is continuous (Nonlinear systems, Khalil).
My question: are there any additional conditions needing to be added if we want to show that the solution $x(t)$ is continuously differentiable on the infinite interval $t \in [0, +\infty)$?
My guess is: we need to add the boundedness or the convergence of the solution $x(t)$. But I don't know how to verify/prove it.
I appreciate any help or suggestion!