Existence of solution to differential equation for all time.

Question

Suppose that the function $f(t,x)$ is bounded and continuous on $\mathbb R \times \mathbb R$ and has continuous partial derivatives with respect to $x$. Prove that any solution $x=\Phi(t)$ of the equation $\dot x=f(t,x)$ is defined for all $t\in (-\infty,\infty)$.

Answer 1

Hint: Suppose $\phi$ is a solution with maximal domain of definition $(a,b)$ and that $b<+\infty$. What can you say about the behaviour of $\phi(t)$ as $t$ approaches $b$?