Assume $U = \mathbb{R} × \mathbb{R}^n$ and that $|f(t,x)| ≤ g(|x|)$ for some continuous function $g \in C([0, \infty))$ satisfying
$\displaystyle\int_0^\infty \dfrac {1}{g(r)}dr = \infty$.
Prove that the solutions of IVP $x' = f(x; t); x(t_0) = x_0$ are defined for all $t ≥ 0$.
Could someone give me a hint how to solve this problem?