Can a Differential Equation Solution Be Extended Indefinitely If Bounded?

[simo1]

if x'=f(x) and x is a solution of that DE on an open interval (alpha,beta)
where f is locally lip.
how can i show that for every possibe extension of x to an interval (apha, tau) the range of the solution is contained in closed and bounded subset of D, then x can be extended to (alpha,infinity

 

[gtoguy97]

)This can be shown by using the definition of locally Lipschitz continuity. A function is locally Lipschitz continuous if it has a finite and non-zero derivative at every point in its domain. This implies that for any x in (α,β), the range of the solution is bounded by the absolute value of the derivative of f(x). Since this bound is independent of τ, it follows that the range of the solution is contained in a closed and bounded subset of D for any possible extension of x to (α,τ). Thus, x can be extended to (α,∞).