Let $U\in\mathbb{R}^d$ be open and let $F\in C^1(U, \mathbb{R}^d)$. Fix $p\in U$. Consider the IVP $$ \dot{x}=F(x),\ x(0)=p $$ Let $(T_-, T_+)$ be the maximal interval be the maximal interval of existence. Assume $$ \lim_{t\uparrow T_+}x(t)=\zeta\in U $$ Prove that $F(\zeta)=0$.
I just learned the extensibility of solution and have difficulty in working out the answer. However, by the extensibility theory, it seems $T_+$ has to be $+\infty$. Do I mistaken something? Please give a concrete explaination.
Any hint or answer of the original question could be very helpful!