The solution of the IVP $\dot{x}=F(t,x),\ x(0)=x_0$ exists on $(-\infty, +\infty)$

Question

Let $F\in C^1$ be bounded. Then the solution of the IVP $$\dot{x}=F(t,x),\ x(0)=x_0$$ exists on $(-\infty, +\infty)$.

I have no idea how to prove the proposition. The $F\in C^1$ condition seems strange and I do not know how to use it.

Answer 1

As $|F(t,x)|\le M$ for some $M<\infty$, you can conclude that for any solution $|y(t)-y_0|\le M\,|t-t_0|$ in its domain. As that gives finite values at all finite times, the solution can be extended without restriction, the maximal solution has $\Bbb R$ as domain.