Consider the following IVP: 
$x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$.

Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$.

In order for the solution to the IVP to exist for all $t>=0$, what conditions should we impose on $b(t)$ and $x_0$?

I know that if the function $f: \mathbb{R^{n+1}}\to \mathbb{R^n}$ is continuous, then there is a theorem (Peano Local existence Theorem) that guarantees that the solution to the IVP exists, but I can't seem to find a way to prove the continuity of $f$. I wonder if anyone of you knows another way on how to solve this problem. Thanks!