Equilibrium points and lipschitz functions

[simo1]

Given a function H_n-H_n is localy lip on its domain and f(p)=0 for some p an element of D
if x is a solution of the system x^' =f(x) on interval I and x(s) not equal to p for some s in I.

how can i show that x(t) is not equal to p for all t in I

 

[gtoguy97]

?Since Hn-Hn is locally Lipschitz on its domain and f(p)=0 for some p in D, it follows that there exists a neighborhood of p in which the function f is Lipschitz continuous (see Rademacher's Theorem). This implies that any solution x of the system x' =f(x) on interval I has a Lipschitz constant L>0. Now, assume by contradiction that there exists t∈I such that x(t)=p. Then, by the Lipschitz continuity of the solution, it follows that |x(s)−p|≤L|s−t| for all s∈I. Since x(s)≠p for some s∈I, this would imply that x(s)−p≠0 for some s∈I, contradicting our assumption that x(t)=p. Therefore, we can conclude that x(t)≠p for all t∈I.