Suppose $f(t, x)\in C^1(\mathbb R^2)$ and satisfies $|f(t, x) |\leq 1+|x|$ for any $(t, x) \in\mathbb R^2$. Then prove that the IVP $x'(t) =f(t, x)$ and $x(0)=0$ has a solution defined on whole $\mathbb R$.
I tried to find local solutions by restricting $f$ to rectangles. But it seems like it couldn't be guaranteed that the solution even exists on the whole interval. Another idea is to try to prove it by approximating $f$ uniformly by polynomials. I'm sure how to make it work. I'd like to get hints on how to solve this. Thanks in advance.
You can use the extensibility theorem - see document, Theorem 1.20 and its corollary about maximal solutions to prove what you want. Or mimic the proof to your particular example.